Question

When rolling a die 100 times what is the probability of rolling a 4 exactly 25 times?

Answer 1

Using the binomial distribution, the required probability is given by:
$100C25\times(\frac{1}{6})^{25}\times(\frac{5}{6})^{75}=0.0098$

Answer 2

Using the binomial distribution, it is found that there is a 0.0098 = 0.98% probability of rolling a 4 exactly 25 times.

For each die, there are only two possible outcomes, either a 4 is rolled, or it is not. The result of each roll is independent of other rolls, hence, the binomial distribution is used to solve this question.

Binomial probability distribution

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$C_{n,x} = \frac{n!}{x!(n-x)!}$

The parameters are:

• x is the number of successes.

• n is the number of trials.

• p is the probability of a success on a single trial.

In this problem:

• The die is rolled 100 times, hence $n = 100$.

• There are 6 sides, one of which is 4, hence $p = \frac{1}{6} = 0.1667$

The probability is P(X = 25), hence:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 25) = C_{100,25}.(0.1667)^{25}.(0.8333)^{75} = 0.0098$

0.0098 = 0.98% probability of rolling a 4 exactly 25 times.

For more on the binomial distribution, you can check https://brainly.com/question/24863377