McClass

Question

What is the probability for exactly $2$ 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 $k$ 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 with 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.

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

Correctly recognised the problem as a binomial probability.
Correctly identified the formula for the binomial probability.
Please determine the number of permutations using the binomial coefficient $(\binom{N}{k}) = \frac{N!}{k! \cdot (N - k)!}$ .
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.

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

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.
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.

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

Correctly recognised the problem as a binomial probability.
Correctly identified the formula for the binomial probability.
Please determine the number of permutations using the binomial coefficient $(\binom{N}{k}) = \frac{N!}{k! \cdot (N - k)!}$ .
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.

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

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.

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

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

Traps

Avoid these traps

Do not forget to multiply everything with the number of permutations $(\binom{N}{k})$ .
Make sure not to confuse the probabilities $π$ for success and $1 - π$ for failure.
Make sure not to forget the probability for the $N - k$ failures.
The number of permutations is determined using the binomial coefficient $(\binom{N}{k}) = \frac{N!}{k! \cdot (N - k)!}$ .

Binary outcomes

Question

Result 1

Result 2

0.19

0.00

0.00

0.00

0.00

0.02

0.12

0.58

5.20

141.14

7551.66

209744.64