A
community has m families with children. The largest family has k
children. For i = 1,… , k, there are ni families with i children,
so n1 + n2 + ….
+
nk = m. Choose a child at random from the community in one of the
following ways:
(a)
Choose one of the m families at random, and then choose one of the
children at random from that family.
(b)
Choose one of the n1 + 2n2 + …. + knk children at random.
Now
we have the following questions:
(1)
What is the probability that the child chosen is the first-born in
his/her family if you use method (a)? (Hint. Consider the Total
Probability
Theorem)
(2)
What is the probability that the child chosen is the first-born in
his/her family if you use method (b)?
(3)
Which method results in the larger probability that the child
chosen is the first-born in his/her family? (Hint. Use the
Cauchy-Schwartz
inequality)
I
already got parts (1) and (2), I’m not sure how to apply the
Cauchy-Schwartz inequality to this though.
already got parts (1) and (2), I’m not sure how to apply the
Cauchy-Schwartz inequality to this though.
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