Probability questions can be tricky because they are sensitive to the precise kind of information (i.e. conditioning) available. This problem is taken from Machine Learning: a Probabilistic Perspective by Kevin Murphy. My new neighbors have two children. Let’s assume the sexes of their children are independent from one another, each with probability 1 2 , so that all four possibilities {(B, B),(B, G),(G, B),(G, G)} are equally likely (we are writing the sexes in (oldest, youngest) order). (a) If I ask my neighbors whether they have any boys, and they say “yes,” then what is the probability that one of their two children is a girl? Clearly list out the sample space and associated probabilities. (b) Suppose instead that I happen to see one of their children run by, and it’s a boy. What is the probability that their other child is a girl? (Hint: what is this probability if the child I saw was their oldest child? How about their youngest? How can we synthesize this information to get an overall answer?) Are you surprised that these answers are different? Can you articulate what is different about them which causes the calculation to change?
Probability questions can be tricky because they are sensitive to the precise kind of information (i.e. conditioning) available. This problem is taken from Machine Learning: a Probabilistic Perspective by Kevin Murphy. My new neighbors have two children. Let’s assume the sexes of their children are independent from one another, each with probability 1 2 , so that all four possibilities {(B, B),(B, G),(G, B),(G, G)} are equally likely (we are writing the sexes in (oldest, youngest) order). (a) If I ask my neighbors whether they have any boys, and they say “yes,” then what is the probability that one of their two children is a girl? Clearly list out the
Step by step
Solved in 3 steps with 16 images
