Download 1 - Art of Problem Solving

1. A fair coin is tossed repeatedly. What is the average number of tosses required to obtain a
head?
2. A die is rolled repeatedly. On average, how many times must the die be rolled to get a 6?
3. A couple decides to keep having children until they have at least one boy and one girl. What
is the average number of children they will have? (Assume no twins.)
4. I alternately toss a fair coin and throw a fair die until I either toss a head or throw a 2. If I
toss the coin first, what is the probability that I throw a 2 before I toss a head?
5. A fair coin is tossed repeatedly. What is the average number of tosses required to obtain
two heads in a row?
6. An apple is located at vertex
of pentagon
, and a worm is located two vertices
away, at . Every day the worm crawls with equal probability to one of the two adjacent
vertices. When it reaches vertex , it stops to dine. What is the average number of days until
dinner?
7. Five people stand at the vertices of a pentagon, throwing frisbees at each other. They have
two frisbees, initially at adjacent vertices. At each step, each frisbee is thrown either to the
left or to the right with equal probability. This process continues until one person is the
target of two frisbees simultaneously; then the game stops. What is the average number of
steps taken?
8. Three men, , , and , are to fight a pistol duel (truel?). 's chance of hitting a target
is
, 's chance is
, and
never misses. They are to fire at their choice of target in
succession in the order , , , cyclically (but a hit man loses further turns and is no longer
shot at) until one man is left. What should 's strategy be?
9. Erica flips a fair coin. Nate then flips the coin and wins if it matches Erica's flip. If it doesn't
match, then Noah flips the coin and wins if it matches Nate's flip. If Noah doesn't win, then
Erica flips and wins if it matches Noah's flip. The game continues until there is a winner. What
is the probability that Nate wins the game?
10. Two persons,
and , play with a die an unlimited number of times.
wins a game if
the outcome is 1 or 2; wins the game in the other cases. A player wins the match if he wins
two consecutive games. Determine the probability that
wins the match.
11. Daniel and Scott are playing a game where a player wins as soon as he has two points
more than his opponent. Both players start at zero, and points are earned one at a time. If
Daniel has a
chance of winning each point, what is the probability that he will win the
game?
12. Coupons are each randomly numbered 1 to 5. All five numbers are required to win a prize.
On average, how many coupons must be taken to win?
13. Alice and Bob flip a coin until either HHT or HTT occurs. Alice wins if the pattern HHT
comes first; Bob wins if HTT comes first. What is the probability that Alice will win?
14. A fair coin is tossed repeatedly. What is the probability that the first occurrence of three
heads in a row is earlier than the first occurrence of two tails in a row?
15. A fair coin is tossed repeatedly. What is the probability that the first occurrence of HH
will occur before the first occurrence of THTH?
16. A fair coin is tossed repeatedly. What is the average number of tosses required to obtain
the sequence THTTH?
17. Player
has 1 dollar, and Player
has 2 dollars. Each play gives one of the players 1
dollar from the other. Player
is enough better than Player
They play until one is bankrupt. What is the chance that Player
that he wins
wins?
of the plays.
18. Two players,
and , start with
and
dollars, respectively. A fair coin is repeatedly
tossed. If the result is heads, then
pays
one dollar. Otherwise,
pays
one dollar. The
game ends when one player goes broke. What is the probability that
wins?
19. A bug is on a number line, starting at 0. A fair coin is tossed repeatedly. If the coin comes
up heads, the bug goes 1 unit left; otherwise, the bug goes 1 unit right. What is the
probability that the bug eventually returns to 0?
20. From where the drunken man stands, one step toward the cliff would send him over the
edge. He takes random steps, either toward or away from the cliff. At any step, his
probability of taking a step away is
falling over the edge?
, of a step toward the cliff
. What is his chance of
21. A microbe either splits into two perfect copies of itself or dies. If the probability of
splitting is (and is independent of the microbe's ancestry), what is the probability that a
microbes's descendants die out?
22. Two players,
and , repeatedly play a game of chance. At each turn, either wins 1
dollar from B with probability
, or
loses 1 dollar to
with probability
. Both
and
have unlimited capital. What is 's expected peak cumulative loss?
23. A particle moves at each step 2 units to the right or 1 unit to the left, with probability
each. The particle starts at 1. What is the probability that the particle will ever reach the
origin?
24. A coin is tossed repeatedly. What is the probability that at some point, the number of
tails will be greater than twice the number of heads?
25. Real numbers are chosen at random from the unit interval (
). If after choosing
the th number the sum of the numbers so chosen first exceeds 1, what is the expected value
of ?
26. Real numbers are chosen at random from the unit interval (
). If after choosing
the th number the sum of the numbers so chosen first exceeds 2, what is the expected value
of ?