Question 1026682: 25% of all classmates find this puzzle difficult. You select four random classmates. What is the probability that exactly two of them find this puzzle difficult?
Answer by mathmate(429)   (Show Source):
You can put this solution on YOUR website!
Question:
25% of all classmates find this puzzle difficult. You select four random classmates. What is the probability that exactly two of them find this puzzle difficult?
Solution:
This is not exactly an ideal application of the binomial distribution because the size of a class is generally between 25 and 40, or in university, it can go up to the hundreds. Since the size of the class is not given/available, we will approximate the solution assuming a large class size, hence the probability of randomly picking a student who finds the puzzle difficult remains constant at 25$.
Then we will apply the binomial distribution, with 4 Bernoulli trials (outcome is either yes or no). We need the probability of finding exactly two out of the four students find the puzzle difficult.
The probability is given by:
P(X≤k)=∑ P(X=i) for i=0,1,2...k.
where
P(X=i)=C(n,i)*p^i*(1-p)^(n-i)
and
C(n,i)=n!/(i!(n-i)!)
n=number of trials
i=number of successes
p=probability of success.
For this particular problem,
n=4,
k=2,
p=0.25
so
P(X≤k)=∑ P(X=i) for i=0,1,2
=P(X=0)+P(X=1)+P(X=2)
=C(4,0)*0.25^0*(1-0.25)^4+C(4,1)*0.25^1*(1-0.25)^3+C(4,2)*0.25^2*(1-0.25)^2
=0.31641+0.42188+-/21094
=0.94922
Answer: The probability of up to two out of four students finding the puzzle difficult is approximately 0.94922.
(read paragraph regarding assumptions)
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