Question

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 ball. (a) Alice and Bob decide to play this game with Alice being first to draw a ball from the bucket. What is the probability that Alice wins the game. (b) Bob thinks it is unfair that Alice always goes first in these games. Suppose they decide who goes first based on the toss a fair coin. If the result of the coin toss in heads, then Alice plays first. Otherwise, Bob plays first. What is the probability that Alice wins? (c) Assume that n = 10. What is the probability that 10 or more balls are drawn before there is a winner?

Answer #1

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.

Box A contains 5 white balls and 7 black balls. Randomly choose
one ball. If it is white ball, we randomly draw one more from box
A, and if it is black, we choose one ball from box B, in which
there are 7 white balls and 8 black balls.
a) What is the probability that in the second draw we will
choose a white ball?
b) The second ball is white, what is the probability that the
first one...

You play the following game against your friend. You have 2 urns
and 4 balls One of the balls is black and the other 3 are white.
You can place the balls in the urns any way that you'd like,
including leaving an urn empty. Your friend will choose one urn at
random and then draw a ball from that urn. ( If he chooses an empty
urn, he draws nothing.) She wins if she draws the black ball and...

2. One white ball, one black ball, and two yellow balls are
placed in a bucket. Two balls are drawn simultaneously from the
bucket. You are given the information that at least one of them is
yellow. What is the probability that the second ball is also
yellow?
3. Students taking the GMAT were asked about their undergraduate
major and intent to pursue their MBA as a full-time or part-time
student. A summary of their responses follows. a. If a...

A box contains one white ball, two red balls, and three black
balls. Make a box model.
Five draws are made with replacement from the
box. Find the chance that:
a) A red ball is never drawn.
b) A black ball appears exactly three
times.
c) A white ball appears at least once.

An urn contains 20 balls, 5 of which are white. Three players -
A, B, and C - successively draw from the
urn. Each ball is replaced after it is drawn. The winner is the
first one to draw a white ball.
(a)
Find the probability that A wins.
(b)
Find the probability that C wins
Please explain steps. Thank you :)

Suppose that a person plays a game in which he draws a ball from
a box of 6 balls numbered 1 through 6. He then puts he ball back
and continues to draw a ball (with replacement) until he draws
another number which is equal or higher than the first draw. Let X
and Y denote the number drawn in the first and last try.
respectively.
a) Find the probability distribution of X (The first draw)
b) Find the probability...

A box contains 4 white balls and 4 black balls, and the
following game is played. Each round of the game consists of
randomly selecting four balls from the box. If exactly two of them
are white, the game is over. Otherwise, the four balls are placed
back in the box, so that it again contains 4 white balls and 4
black balls before the next round is performed. The rounds are
repeated until exactly two of the chosen balls...

Given 3 urns, one contains 2 black balls, the second contains 2
white balls, and the third contains 1 ball of each color. An
urn is chosen at random and a ball is drawn from it -- the ball is
white. What is the probability that the other ball in that urn is
also white?

An urn contains 4 white marbles and 1 black marble. Alex, Ben,
and Cole take turns drawing a marble from the urn with replacement,
in the order Alex, then Ben, then Cole. The first person to draw
the black marble is eliminated from the game.
a) Find the probability that Ben is eliminated first.
b) After the first person is eliminated, the game continues with
the two remaining players. Find the probability that Alex wins the
whole game.

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