Question

A probability class has 30 students. As part of an assignment, each student tosses a coin 200 times and records the number of heads. Approximately what is the chance that no student gets exactly 100 heads?

Answer 1

17.58% probability that no student gets exactly 100 heads

Explanation:

We use the binomial probability distribution twice 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.

Probability of a single student getting 100 heads.

The coin is tossed 200 times, so
`n = 100`

For each toss, 50% probability of getting heads, so
`p = 0.5`

Then

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

`P(X = 100) = C_(200,100).(0.5)^(100).(0.5)^(100) = 0.0563`

Approximately what is the chance that no student gets exactly 100 heads?

Each student has a 5.63% probability of getting exactly 100 heads, so
`p = 0.0563`

30 students, so
`n = 30`

We have to find P(X = 0).

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

`P(X = 0) = C_(30,0).(0.0563)^(0).(0.9437)^(30) = 0.1758`

17.58% probability that no student gets exactly 100 heads

Answer 2

The probability that a student gets exactly 100 heads out of 200

We have to combination(P&C) of 200 with 100 (200C100)/2^200

= (200!)/(100! * 100! * 2^200) = P

The probability that the student doesn't get exactly 100 heads out of 200 = 1-P

The probability that all 30 students can't get exactly 100 heads out of 200 = (1-P)^30

[1- (200!)/(100! * 100! * 2^200)]^30

= 17.45% approximately.

This is the required solution.