Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 851 — #859
19.5. Linearity of Expectation 851
(c) What is his expected final profit (amount won minus amount lost)?
(d) You can beat a biased game by bet doubling, but bet doubling is not feasible
because it requires an infinite bankroll. Verify this by proving that the expected size
of the gambler’s last bet is infinite.
Problem 19.36.
Six pairs of cards with ranks 1–6 are shuffled and laid out in a row, for example,
1 2 3 3 4 6 1 4 5 5 2 6
In this case, there are two adjacent pairs with the same value, the two 3’s and the
two 5’s. What is the expected number of adjacent pairs with the same value?
Problem 19.37.
There are six kinds of cards, three of each kind, for a total of eighteen cards. The
cards are randonly shuffled and laid out in a row, for example,
1 2 5 5 5 1 4 6 2 6 6 2 1 4 3 3 3 4
In this case, there are two adjacent triples of the same kind, the three 3’s and the
three 5’s.
(a) Derive a formula for the probability that the 4th, 5th, and 6th consecutive cards
will be the same kind—that is, all 1’s or all 2’s or. . . all 6’s?
(b) Let p WWD PrŒ4th, 5th and 6th cards match—that is, p is the correct answer to
part (a). Write a simple formula for the expected number of matching triples in
terms of p.