Question

Timothy creates a game in which the player rolls 4 dice. What is the probability in this game of having exactly two dice or more land on a five?

A. 0.016
B. 0.132
C. 0.868
D. 0.984

Answer 1

B. 0.132

Explanation:

For each time the dice is thrown, there are only two possible outcomes. Either it lands on a five, or it does not. The probability of a throw landing on a five is independent of other throws. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

Timothy creates a game in which the player rolls 4 dice.

This means that
`n = 4`

The dice can land in 6 numbers, one of which is 5.

This means that
`p = (1)/(6)`

What is the probability in this game of having exactly two dice or more land on a five?

`P(X \geq 2) = P(X = 2) + P(X = 3) + P(X = 4)`

In which

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 2) = C_(4,2).((1)/(6))^(2).((5)/(6))^(2) = 0.116`

`P(X = 3) = C_(4,2).((1)/(6))^(3).((5)/(6))^(1) = 0.015`

`P(X = 4) = C_(4,4).((1)/(6))^(4).((5)/(6))^(0) = 0.001`

`P(X \geq 2) = P(X = 2) + P(X = 3) + P(X = 4) = 0.116 + 0.015 + 0.001 = 0.132`

So the correct answer is:

B. 0.132