A person randomly picks 3 out of 10 gold coins without repla

A person randomly picks 3 out of 10 gold coins without replacement and each gold coin might be a counterfiet with a probability of 0.2.

a) What is the probability that the coins will be counterfeits?

b) What is the probability that at most two of the coins will be counterfeits?

c) How many counterfiets can this person expect?

I think I understand A and B with my solutions below but am having trouble understanding how to approach C.

a) P(X=3) = (0.2)^3 = 0.08

b)P(X=2) = (0.2)^2 * 0.8; P(X=1) = 0.2 * (0.8)^2; P(X=0) = (0.8)^3; P(X<=2) = P(X=2) + P(X=1) + P(X=0) = 0.672

Solution

We have given that, A person randomly picks 3 out of 10 gold coins without replacement.

Let X be the random variable that the number of gold coin might be a counterfiet.

Given that P( gold coin might be a counterfiet) = 0.2

It means p = 0.2 , q = 1 - p = 0.8

Here we use Hypergeometric distribution.

We use Hypergeometric distribution when,

i) Sample size n is randomly drawn from a population of size N without replacement.

ii)It is known that there are k successes in the population and thus there are N - k failuares in the population

a) What is the probability that the coins will be counterfeits?

We have given that

P( gold coin might be a counterfiet) = 0.2

And we have to calculate P(all the three coins will be counterfiets) = 0.2 ^ 3 =0.008

b) What is the probability that at most two of the coins will be counterfeits?

That is we have to calculate P(X 2 ).

P(X 2 ) = P(X = 0) + P(X = 1) + P(X = 2)

The p.m.f. of Hypergeometric distribution is,

P(X = x) = {( k C x) * [ ( N - k) C (n-x) ] } / (N C n) ( C is used for combination)

Here N = 10 , n = 3 , p = 0.2 , k = 3.

We calculate first probabilities.

P(X = 0) = [(3 C 0) * (7 C 3) ] /(10 C 3) = 0.29167

P(X = 1) = [(3 C 1) * (7 C 2) ] /(10 C 3) = 0.525

P(X = 2) = [(3 C 2) * (7 C 1) ] /(10 C 3) = 0.175

P(X 2) = 0.29167 + 0.525 + 0.175 = 0.99167

c) How many counterfiets can this person expect?

There are 10 gold coins out of 10 person randomly picks 3 coins without replacement.

10 counterfiets can this person expect because we have to select 3 coins these three coins are whichever in between 10 either 1st,2nd,3rd,.......,10th.

We don\'t know which coin is picked up in person\'s hand.

that\'s why the person may be select out of 10 coins.