Problem 5: (10 points) A couple decided they will continue to have children unti

Question

Problem 5: (10 points) A couple decided they will continue to have children until they have two males. Assume that P(male) 0.5. (a) What is the probability that their second male is their sixth child? (b) How many children would the couple expect to have in order to have two males? o) Find the standard deviation of the nmnber of children that couple would have until they have two males Problem 6: (20 points) Suppose that airplanes arrive at a certain airport with rate ? 2 per minute. (a) What's the probablity exactly three planes arrive during a 1-min period? (b) Wh period? (c) What's the probability that exactly five planes arrive during a 2-min pcriod? (d) What is the mean value of the mumber of planes thnt arrive during a half hour period? at's the probability that at least two airplanes arrive during a 1-min

Explanation / Answer

P(Male)=0.5

a) _ _ _ _ _ M

1 2 3 4 5 6

out of these 5 we have to arrange 4 female and 1 male and hence the total ways are = 5C4 =5

now each point has the probability of 0.5 (as P(Female)=1-P(Male) = 1-0.5 =0.5, so P(Male)=P(Female)=0.5)

So ans = 5*0.5^6 = 0.078125

0.078125 is the prob of having their second child male as 6th child

b) The couple could end up with the following situations

(i)MM

(ii)MFM

(iii)MFFM ....

ans= 2*0.5*0.5 + 3*0.5*0.5*0.5+4*0.5*0.5*0.5*0.5+-----

= 2*0.5*0.5/(1-0.5) + 0.5*0.5*0.5/(1-0.5) + ---

= 2*0.5+0.5*0.5+---

= 1+0.5/(1-0.5)

=2

So., on an avg a couple would have 2 children

C) the possible number of children would start from 2,3,4,---,inf

so the std = 1 (the gap between each number)

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