Question

A die is rolled 20 times and the number of twos that come up is tallied. Find the probability of getting the given result. [Binomail Probability] Less than four twos

Answer 1

0.5665 = 56.65% probability of less than four twos.

Explanation:

For each roll, there are only two possible outcomes. Either it is a two, or it is not a two. The probability of a roll ending up in a two is independent of any other roll, which means that the binomial probability distribution is used.

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.

A die is rolled 20 times

This means that
`n = 20`

One out of six sides is 2:

This means that
`p = (1)/(6) = 0.1667`

Probability of less than four twos:

This is:

`P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)`

So

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

`P(X = 0) = C_(20,0).(0.1667)^(0).(0.8333)^(20) = 0.0261`

`P(X = 1) = C_(20,1).(0.1667)^(1).(0.8333)^(19) = 0.1043`

`P(X = 2) = C_(20,2).(0.1667)^(2).(0.8333)^(18) = 0.1982`

`P(X = 3) = C_(20,3).(0.1667)^(3).(0.8333)^(17) = 0.2379`

So

`P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.0261 + 0.1043 + 0.1982 + 0.2379 = 0.5665`

0.5665 = 56.65% probability of less than four twos.