Here is another basic rule of probability: if several events are independent, th

Question

Here is another basic rule of probability: if several events are independent, the probability that all of the events happen is the product of their individual probabilities. We know, for example, that a child has probability 0.49 of being a girl and probability 0.51 of being a boy and that the sexes of successive children are independent. So the probability that a couple's two children are two girls is 0.49 0.49 = 0.2401. You can use this multiplication rule to calculate the probability that a coupe who plan to have children until they have gr o n the have three ch dren, whichever comes first, will have a girl among their children. (a) Consider all eight possible arrangements of the sexes of three children, for example, BBB and BBG. Use the multiplication rule to find the probability of each outcome. Check your work by verifying that your eight probabilities add to 1. BBG BGB GBB GGB GBG BGG (b) A couple plan to stop when they have a girl or to stop at three children even if all are boys. Use your work from part (a) to find the probability that they get a girl.

Explanation / Answer

a) The required probabilities here are computed as:

b) Probability that the family has a girl is computed in the following cases as:

P(G) + P(BG) + P(BBG) = 0.49 + 0.51*0.49 + 0.512*0.49 = 0.867349

P(BBB) = 0.513 = 0.132651

Therefore probability that they get a girl is computed as:

= 0.867349 / (0.867349 + 0.132651) = 0.867349

Therefore 0.867349 is the required probability here.

OUTCOME Probability BBB 0.513 = 0.132651 BBG 0.512*0.49 = 0.127449 BGB 0.512*0.49 = 0.127449 GBB 0.512*0.49 = 0.127449 GGB 0.492*0.51 = 0.122451 GBG 0.492*0.51 = 0.122451 BGG 0.492*0.51 = 0.122451 GGG 0.493 = 0.117649

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