a TaySachs disease is a rare but fatal genetic disease If a

(a) Tay-Sachs disease is a rare but fatal genetic disease. If a couple are both carriers, then
a child born to them has a probability 0.25 of having the disease. If such a couple
have 5 children, what is the probability that two or more of the children will have the
disease?

(b) A piece of electronic equipment contains eight transistors. This piece of equipment
is to be tested: the testing regime involves selecting two transistors at random and
checking to see if they are defective. The equipment only passes the test if neither of
the randomly selected transistors are defective. If the piece of equipment has three
defective transistors, what is the probability that it will pass the test?

Solution

(a) Given X~Binomial(n=5, p=0.25)

P(X=x) =5Cx*(0.25^x)*(0.75^(5-x))

So the probability that two or more of the children will have the disease is

P(X>=2) =1-P(X=0) - P(X=1)

=1- 5C0*(0.25^0)*(0.75^(5-0)) - 5C1*(0.25^1)*(0.75^(5-1))

=0.3671875

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(b) Number of defective transistors is 3

Number of non defective transistors is 5

the probability that it will pass the test is

P(pass the test | three defective transistors)

=5C2 / 8C2

= (5*4/2) / (8*7/2)

=0.3571429