Question

Alice and Bob play the following game. They toss 5 fair coins. If all tosses are Heads,Bob wins. If the number of Heads tosses is zero or one, Alice wins. Otherwise they repeat,tossing five coins on each round, until the game is decided. (a) Compute the expectednumber of coin tosses needed to decide the game. (b) Compute the probability that Alicewins.

Answer #1

Alice and Bob play a game in which they flip a coin repeatedly.
Each time the coin is heads, Alice wins $1 (and Bob loses $1). Each
time the coin is tails, Bob wins (and Alice loses) $2. They
continue playing until Alice has won three flips. Prove that the
expected value of Bob’s winnings is $3. (Hint: Use linearity of
expected value to consider the expected value of each flip
separately, with flips being worth $0 if they do...

Alice and Bob have 9 coins, each with probability of a head
equal to p = .6. Bob tosses 5 coins, while Alice tosses the
remaining 4 coins. Assuming that all tosses are independent,
compute the probability that Bob gets more heads than Alice.

Alan tosses a coin 20 times. Bob pays Alan $1 if the first toss
falls heads, $2 if the first toss falls tails and the second heads,
$4 if the first two tosses both fall tails and the third heads, $8
if the first three tosses fall tails and the fourth heads, etc. If
the game is to be fair, how much should Alan pay Bob for the right
to play the game?

3. Consider the following game. A bucket contains one black ball
and n − 1 white balls. Two players take it in turn to draw balls
from the bucket. On each turn, a player draws a ball from the
bucket. If the player draws the black ball, then that player wins
and the game ends. If the player draws a white ball, then the ball
is returned to the bucket and the game continues until one player
draws the black...

We toss n coins and each one shows up heads with probability p,
independent of the other coin tosses. Each coin which shows up
heads is tossed again.
What is the probability mass function of the number of heads
obtained after the second round of coin tossing?

David and Carol play a game as follows: David throws a die, and
Carol tosses a coin. If die falls "six", David wins. If the die
does not fall "six" and the coin does fall heads, Carol wins. If
neither the die falls "six" nor the coin falls heads, the foregoing
is to be repeated as many times as necessary to determine a winner.
Whant is the probability that David wins?
Answer is 2/7

Amanda, Becky, and Charise toss a coin in sequence until one
person “wins” by tossing the first head. a) If the coin is fair,
find the probability that Amanda wins. b) If the coin is fair, find
the probability that Becky wins. c) If the coin is not necessarily
fair, but has a probability p of coming up heads, find an
expression involving p for the probability that Becky wins. d) As
in part c) find similar expressions for Amanda...

I toss 3 fair coins, and then re-toss all the ones that come up
tails. Let X denote the number of coins that come up heads on the
first toss, and let Y denote the number of re-tossed coins that
come up heads on the second toss. (Hence 0 ≤ X ≤ 3 and 0 ≤ Y ≤ 3 −
X.)
(a) Determine the joint pmf of X and Y , and use it to calculate
E(X + Y )....

1. Let X be the number of heads in 4 tosses of a fair coin.
(a) What is the probability distribution of X? Please show how
probability is calculated.
(b) What are the mean and variance of X?
(c) Consider a game where you win $5 for every head but lose $3
for every tail that appears in 4 tosses of a fair coin. Let the
variable Y denote the winnings from this game. Formulate the
probability distribution of Y...

There is a game with two players. Both players place $1 in the
pot to play. There are seven rounds and each round a fair coin is
flipped. If the coin is heads, Player 1 wins the round. Otherwise,
if it is tails, Player 2 wins the round. Whichever player wins four
rounds first gets the $2 in the pot.
After four rounds, Player 1 has won 3 rounds and Player 2 has
won 1 round, but they cannot finish...

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