McClass

Question

What is the probability of exactly $8$ successes?

This is a binomial probability.
The formula for the binomial probability is $(\binom{N}{k}) \cdot π^{k} \cdot {(1 - π)}^{N - k}$ .
Determine the probability for a sequence of $k$ successes and $N - k$ failures.
Determine the probability for $k$ independent successes.
Determine the probability for $k$ independent failures.

This is a binomial probability.
The formula for the binomial probability is $(\binom{N}{k}) \cdot π^{k} \cdot {(1 - π)}^{N - k}$ .
Determine the probability for a sequence of $k$ successes and $N - k$ failures.
Determine the probability for $k$ independent failures.
Determine the probability for $k$ independent successes.

Feedback

Waiting for response...

Solution(s)

The system accepts the following correct response(s)

$(\binom{N}{k}) \cdot π^{k} \cdot {(1 - π)}^{N - k}$

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.

$(\binom{N}{k}) \cdot π^{k} \cdot {(1 - π)}^{N - k}$

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.

Incorrect responses

The system detects the following incorrect responses

$(\binom{N}{k}) \cdot \underset{instead of π^{k}}{\underset{⏟}{{(1 - π)}^{k}}} \cdot \underset{instead of {(1 - π)}^{N - k}}{\underset{⏟}{π^{N - k}}}$

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

$(\binom{N}{k}) \cdot \underset{instead of π^{k}}{\underset{⏟}{{(1 - π)}^{k}}} \cdot \underset{instead of {(1 - π)}^{N - k}}{\underset{⏟}{π^{N - k}}}$

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

$(\binom{N}{k}) \cdot π^{k} \cdot {(1 - π)}^{N - k}$

Correctly recognised the problem as a binomial probability.
The binomial coefficient for 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.

$(\binom{N}{k}) \cdot π^{k} \cdot {(1 - π)}^{N - k}$

Correctly recognised the problem as a binomial probability.
The binomial coefficient for 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.

$(\binom{N}{k}) \cdot π^{k} \cdot {(1 - π)}^{N - k}$

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

$(\binom{N}{k}) \cdot \underset{instead of π^{k}}{\underset{⏟}{{(1 - π)}^{k}}} \cdot \underset{instead of {(1 - π)}^{N - k}}{\underset{⏟}{π^{N - k}}}$

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

$(\binom{N}{k}) \cdot \underset{instead of π^{k}}{\underset{⏟}{{(1 - π)}^{k}}} \cdot \underset{instead of {(1 - π)}^{N - k}}{\underset{⏟}{π^{N - k}}}$

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

$(\binom{N}{k}) \cdot π^{k} \cdot {(1 - π)}^{N - k}$

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.

$\underset{instead of (\binom{N}{k})}{\underset{⏟}{\frac{N}{k}}} \cdot \underset{instead of π^{k}}{\underset{⏟}{{(1 - π)}^{k}}} \cdot \underset{instead of {(1 - π)}^{N - k}}{\underset{⏟}{π^{N - k}}}$

Correctly recognised the problem as a binomial probability.
Correctly identified the formula for the binomial probability.
The number of permutations was mistakenly calculated by dividing the number of trials $N$ by the number of successes $k .$
The probabilities $π$ for success and $1 - π$ for failure were confused.
Correctly determined the probability for $N - k$ independent failures.
Correctly determined the probability for $k$ independent successes.

Correctly recognised the problem as a binomial probability.
Correctly identified the formula for the binomial probability.
The number of permutations was mistakenly calculated by dividing the number of trials $N$ by the number of successes $k .$
The probabilities $π$ for success and $1 - π$ for failure were confused.
Correctly determined the probability for $k$ independent successes.
Correctly determined the probability for $N - k$ independent failures.

$\underset{instead of (\binom{N}{k})}{\underset{⏟}{\frac{N}{k}}} \cdot π^{k} \cdot {(1 - π)}^{N - k}$

Correctly recognised the problem as a binomial probability.
Correctly identified the formula for the binomial probability.
The number of permutations was mistakenly calculated by dividing the number of trials $N$ by the number of successes $k .$
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.

$\underset{instead of (\binom{N}{k})}{\underset{⏟}{\frac{N}{k}}} \cdot π^{k} \cdot {(1 - π)}^{N - k}$

Correctly recognised the problem as a binomial probability.
Correctly identified the formula for the binomial probability.
The number of permutations was mistakenly calculated by dividing the number of trials $N$ by the number of successes $k .$
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.

$\underset{instead of (\binom{N}{k})}{\underset{⏟}{\frac{N}{k}}} \cdot π^{k} \cdot {(1 - π)}^{N - k}$

Correctly recognised the problem as a binomial probability.
Correctly identified the formula for the binomial probability.
The number of permutations was mistakenly calculated by dividing the number of trials $N$ by the number of successes $k .$
The probability for the $N - k$ failures was omitted.
Correctly determined the probability for $k$ independent successes.

$\underset{instead of (\binom{N}{k})}{\underset{⏟}{\frac{N!}{k!}}} \cdot \underset{instead of π^{k}}{\underset{⏟}{{(1 - π)}^{k}}} \cdot \underset{instead of {(1 - π)}^{N - k}}{\underset{⏟}{π^{N - k}}}$

Correctly recognised the problem as a binomial probability.
Correctly identified the formula for the binomial probability.
$(N - k)!$ is missing in the formula for the binomial coefficient.
The probabilities $π$ for success and $1 - π$ for failure were confused.
Correctly determined the probability for $N - k$ independent failures.
Correctly determined the probability for $k$ independent successes.

Correctly recognised the problem as a binomial probability.
Correctly identified the formula for the binomial probability.
$(N - k)!$ is missing in the formula for the binomial coefficient.
The probabilities $π$ for success and $1 - π$ for failure were confused.
Correctly determined the probability for $k$ independent successes.
Correctly determined the probability for $N - k$ independent failures.

$\underset{instead of (\binom{N}{k})}{\underset{⏟}{\frac{N!}{k!}}} \cdot π^{k} \cdot {(1 - π)}^{N - k}$

Correctly recognised the problem as a binomial probability.
Correctly identified the formula for the binomial probability.
$(N - k)!$ is missing in the formula for the binomial coefficient.
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.

$\underset{instead of (\binom{N}{k})}{\underset{⏟}{\frac{N!}{k!}}} \cdot π^{k} \cdot {(1 - π)}^{N - k}$

Correctly recognised the problem as a binomial probability.
Correctly identified the formula for the binomial probability.
$(N - k)!$ is missing in the formula for the binomial coefficient.
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.

$\underset{instead of (\binom{N}{k})}{\underset{⏟}{\frac{N!}{k!}}} \cdot π^{k} \cdot {(1 - π)}^{N - k}$

Correctly recognised the problem as a binomial probability.
Correctly identified the formula for the binomial probability.
$(N - k)!$ is missing in the formula for the binomial coefficient.
The probability for the $N - k$ failures was omitted.
Correctly determined the probability for $k$ independent successes.

Traps

Avoid these traps

Do not forget $(N - k)!$ in the formula for the binomial coefficient.
The binomial coefficient includes factorials.
Do not forget to multiply by the number of permutations $(\binom{N}{k})$ .
Do not forget to include the probability for the $N - k$ failures.
Do not confuse the probability of success $π$ with the probability of failure $1 - π$

Binary outcomes

Question

Result 1

Result 2

0.08

0.08

0.00

0.00

0.00

0.00

0.01

0.02

0.02

0.44

0.00...0.00

0.00...0.00

0.00...0.00

0.00...0.00

0.01...0.01

0.04...0.04

0.04...0.04

0.17...0.17

0.17...0.17

0.87...0.87