Binary outcomes

$\left(\genfrac{}{}{0ex}{}{N}{k}\right)\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Correctly determined the probability for $N-k$ independent failures.
• Correctly determined the probability for $k$ independent successes.

$\left(\genfrac{}{}{0ex}{}{N}{k}\right)\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Correctly determined the probability for $k$ independent successes.
• Correctly determined the probability for $N-k$ independent failures.

$\cancel{\left(\genfrac{}{}{0ex}{}{N}{k}\right)\cdot }\underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• The binomial coefficient with the number of permutations was omitted.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Correctly determined the probability for $N-k$ independent failures.
• Correctly determined the probability for $k$ independent successes.

$\cancel{\left(\genfrac{}{}{0ex}{}{N}{k}\right)\cdot }\underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• The binomial coefficient with the number of permutations was omitted.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Correctly determined the probability for $k$ independent successes.
• Correctly determined the probability for $N-k$ independent failures.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Correctly determined the probability for $N-k$ independent failures.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Correctly determined the probability for $k$ independent successes.
• Correctly determined the probability for $N-k$ independent failures.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $N-k$ independent failures.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $k$ independent successes.
• Correctly determined the probability for $N-k$ independent failures.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Correctly determined the probability for $N-k$ independent failures.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Correctly determined the probability for $N-k$ independent failures.
• Correctly determined the probability for $k$ independent successes.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Correctly determined the probability for $k$ independent successes.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $N-k$ independent failures.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Correctly determined the probability for $k$ independent successes.
• Correctly determined the probability for $N-k$ independent failures.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.

$\left(\genfrac{}{}{0ex}{}{N}{k}\right)\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Correctly determined the probability for $N-k$ independent failures.
• Correctly determined the probability for $k$ independent successes.

$\left(\genfrac{}{}{0ex}{}{N}{k}\right)\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Correctly determined the probability for $k$ independent successes.
• Correctly determined the probability for $N-k$ independent failures.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Correctly determined the probability for $N-k$ independent failures.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Correctly determined the probability for $k$ independent successes.
• Correctly determined the probability for $N-k$ independent failures.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $N-k$ independent failures.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $k$ independent successes.
• Correctly determined the probability for $N-k$ independent failures.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Correctly determined the probability for $N-k$ independent failures.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Correctly determined the probability for $N-k$ independent failures.
• Correctly determined the probability for $k$ independent successes.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Correctly determined the probability for $k$ independent successes.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $N-k$ independent failures.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot \underset{\text{instead of}{\pi }^k}{\underbrace{{\left(1-\pi \right)}^k}}\cdot \underset{\text{instead of}{\left(1-\pi \right)}^{N-k}}{\underbrace{{\pi }^{N-k}}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probabilities $\pi$ for success and $1-\pi$ for failure were confused.
• Correctly determined the probability for $k$ independent successes.
• Correctly determined the probability for $N-k$ independent failures.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.

$\cancel{\left(\genfrac{}{}{0ex}{}{N}{k}\right)\cdot }{\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• The binomial coefficient with the number of permutations was omitted.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Correctly determined the probability for $N-k$ independent failures.
• Correctly determined the probability for $k$ independent successes.

$\cancel{\left(\genfrac{}{}{0ex}{}{N}{k}\right)\cdot }{\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• The binomial coefficient with the number of permutations was omitted.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Correctly determined the probability for $k$ independent successes.
• Correctly determined the probability for $N-k$ independent failures.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $N-k$ independent failures.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $k$ independent successes.
• Correctly determined the probability for $N-k$ independent failures.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Correctly determined the probability for $N-k$ independent failures.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Correctly determined the probability for $N-k$ independent failures.
• Correctly determined the probability for $k$ independent successes.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Correctly determined the probability for $k$ independent successes.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $N-k$ independent failures.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Correctly determined the probability for $k$ independent successes.
• Correctly determined the probability for $N-k$ independent failures.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Correctly determined the probability for $N-k$ independent failures.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Correctly determined the probability for $k$ independent successes.
• Correctly determined the probability for $N-k$ independent failures.

$\cancel{\left(\genfrac{}{}{0ex}{}{N}{k}\right)\cdot }{\pi }^k\cancel{\cdot {\left(1-\pi \right)}^{N-k}}$

• Correctly recognised the problem as a binomial probability.
• The binomial coefficient with the number of permutations was omitted.
• The probability for the $N-k$ failures was omitted.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot {\pi }^k\cancel{\cdot {\left(1-\pi \right)}^{N-k}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• The probability for the $N-k$ failures was omitted.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot {\pi }^k\cancel{\cdot {\left(1-\pi \right)}^{N-k}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probability for the $N-k$ failures was omitted.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N}{k}}}}\cdot {\pi }^k\cancel{\cdot {\left(1-\pi \right)}^{N-k}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probability for the $N-k$ failures was omitted.
• Correctly determined the probability for $k$ independent successes.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $N-k$ independent failures.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $k$ independent successes.
• Correctly determined the probability for $N-k$ independent failures.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Correctly determined the probability for $N-k$ independent failures.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Correctly determined the probability for $N-k$ independent failures.
• Correctly determined the probability for $k$ independent successes.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Correctly determined the probability for $k$ independent successes.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $N-k$ independent failures.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Correctly determined the probability for $k$ independent successes.
• Correctly determined the probability for $N-k$ independent failures.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Correctly determined the probability for $N-k$ independent failures.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot {\pi }^k\cdot {\left(1-\pi \right)}^{N-k}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for a sequence of $k$ successes and $N-k$ failures.
• Correctly determined the probability for $k$ independent successes.
• Correctly determined the probability for $N-k$ independent failures.

$\left(\genfrac{}{}{0ex}{}{N}{k}\right)\cdot {\pi }^k\cancel{\cdot {\left(1-\pi \right)}^{N-k}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probability for the $N-k$ failures was omitted.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot {\pi }^k\cancel{\cdot {\left(1-\pi \right)}^{N-k}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• The probability for the $N-k$ failures was omitted.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot {\pi }^k\cancel{\cdot {\left(1-\pi \right)}^{N-k}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probability for the $N-k$ failures was omitted.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.
• Correctly determined the probability for $k$ independent successes.

$\underset{\text{instead of}\left(\genfrac{}{}{0ex}{}{N}{k}\right)}{\underbrace{{\displaystyle \frac{N!}{k!}}}}\cdot {\pi }^k\cancel{\cdot {\left(1-\pi \right)}^{N-k}}$

• Correctly recognised the problem as a binomial probability.
• Correctly identified the formula for the binomial probability.
• The probability for the $N-k$ failures was omitted.
• Correctly determined the probability for $k$ independent successes.
• Please determine the number of permutations using the binomial coefficient $\left(\genfrac{}{}{0ex}{}{N}{k}\right)={\displaystyle \frac{N!}{k!\cdot \left(N-k\right)!}}$.