MSTC Mosteller’s Fifty Challenging Problems in
Probability Exam
1. 4, 21 A drawer contains red socks and black socks. When two
socks are drawn at random, the probability that both are
red is 1/2. (a) How small can the number of socks in the
drawer be? (b) How small if the number of black socks is
even?
2. Choose To encourage Elmer's promising tennis career, his father
champion-fa- offers him a prize if he wins (at least) two tennis sets in a
ther-champion row in a three-set series to be played with his father and
with 0.512 rather the club champion alternately: father-champion-father or
than the champion-father-champion, according to Elmer's choice.
father-champi- The champion is a better player than Elmer's father. Which
on-father with series should Elmer choose?
0.384
3. Both the jury A three-man jury has two members each of whom inde-
have the same pendently has probability p of making the correct decision
probability and a third member who flips a coin for each decision
(majority rules). A one-man jury has a probability p of
making the correct decision. Which jury has the better
probability of making the correct decision?
4. 6 On the average, how many times must a die be thrown
until one gets a 6?
5. 1/16, 1/28 In a common carnival game, a player tosses a penny from
a distance of about 5 feet onto the surface of a table ruled
in 1-inch squares. If the penny (3/4 inch in diameter) falls
entirely inside a square, the player receives 5 cents but
does not get his penny back; otherwise, he loses his penny.
If the penny lands on the table, what is his chance to win?
If the 1-inch square is made smaller by thickening the lines
to 1/16 in wide, what is his chance to win?
6. 17/216 or 0.079 Chuck-a-Luck is a gambling game often played at carni-
vals and gabling houses. A player may bet on any one
of the numbers 1, 2, 3, 4, 5, 6. Three dice are rolled. If
the player's number appears on one, two, or three of the
dice, he receives respectively one, two, or three times his
1/
10
, MSTC Mosteller’s Fifty Challenging Problems in
Probability Exam
original stake plus his own money back; otherwise, he
loses his stake. What is the player's expected loss per
unit stake? (Actually, the player may distribute stakes on
several numbers, but each such stake can be regarded as
a separate bet.)
7. Not working, Mr. Brown always bets a dollar on the number 13 at a
Mr. Brown gains roulette against the advice of Kind Friend. To help cure Mr.
+4.68 - 1.89 = Brown of playing roulette, Kind Friend always bets Brown
+2.79 dollars per $20 at even money that Brown will be behind at the end of
36 trials 36 plays. How is the cure working?
(Most American roulette wheels have 38 equally likely
numbers. If the player's number comes up, he is paid 35
times his stake and gets his original stake back; otherwise,
he loses his stake.)
8. 4 × 13!39!/52! We often read of someone wo has been dealt 13 spades
at bridge. With a well-shuffled pack of cards, what is the
chance that you are dealt a perfect hand (13 of one suit)?
(Bridge is played with an ordinary pack of 52 cards, 13 in
each of 4 suits, and each of 4 players is dealt 13.)
9. 0.27071 The game of craps, played with two dice, is one of Amer-
ica's fastest and most popular gambling games. Calculat-
ing the odds associated with it is an instructive exercise.
The rules are these. Only totals for the two dice count. The
player throws the dice and wins at once if the total for the
first throw is 7 or 11, loses at once if it is 2, 3, or 12. Any
other throw is called his "point." If the first throw is a point,
the player throws the dice repeatedly until he either wins
by throwing his point again or loses by throwing 7. What is
the player's chance to win?
10. 2/3 Three prisoners, A, B, and C, with apparently equally good
records have applied for parole. The parole board has
decided to release two of the three, and the prisoners
know this but not which two. A warder friend of prisoner
A knows who are to be released. Prisoner A realizes that
it would be unethical to ask the warder if he, A, is to
2/
10
