Respuesta :
Solution:
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.
Answer:
17.58% probability that no student gets exactly 100 heads
Step-by-step 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.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
Probability of a single student getting 100 heads.
The coin is tossed 200 times, so [tex]n = 100[/tex]
For each toss, 50% probability of getting heads, so [tex]p = 0.5[/tex]
Then
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 100) = C_{200,100}.(0.5)^{100}.(0.5)^{100} = 0.0563[/tex]
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 [tex]p = 0.0563[/tex]
30 students, so [tex]n = 30[/tex]
We have to find P(X = 0).
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{30,0}.(0.0563)^{0}.(0.9437)^{30} = 0.1758[/tex]
17.58% probability that no student gets exactly 100 heads
