Question

An urn contains 4 red balls and 3 green balls. Two balls are sampled randomly.

Let Z denote the number of green balls in the sample when the draws are done without replacement. Give the possible value of Z and its probability mass function (PMF).

Answer #1

An urn contains 1 white, 2 black, 3 red, and 4 green balls. If 6
balls are selected randomly (without replacement) and X represents
the number of selections that are either red or green, find: (a)
the probability mass function for X. (b) the expected value of X
(calculate this value directly by using the probability mass
function from part a).

An urn contains 10 balls, 2 red, 5 blue, and 3 green balls. Take
out 3 balls at a random,
without replacement. You win $2 for each green ball you select and
lose $3 for each red ball you
select. Let the random variable X denote the amount you win,
determine the probability mass
function of X.

Urn A contains 5 green and 4 red balls, and Urn B contains 3
green and 6 red balls. One ball is drawn from Urn A and transferred
to Urn B. Then one ball is drawn from Urn B and transferred to Urn
A. Let X = the number of green balls in Urn A after this process.
List the possible values for X and then find the entire probability
distribution for X.

Urn A contains 6 green and 4 red balls, and Urn B contains 3
green and 7 red balls. One ball is drawn from Urn A and transferred
to Urn B. Then one ball is drawn from Urn B and transferred to Urn
A. Let X = the number of green balls in Urn A after this process.
List the possible values for X and then find the entire probability
distribution for X.

2. Urn A contains 6 green and 4 red balls, and Urn B contains 3
green and 7 red balls. One ball is drawn from Urn A and transferred
to Urn B. Then one ball is drawn from Urn B and transferred to Urn
A. Let X = the number of green balls in Urn A after this process.
List the possible values for X and then find the entire probability
distribution for X.

Urn A contains 5 green and 3 red balls, and Urn B contains 2
green and 6 red balls. One ball is drawn from Urn A and transferred
to Urn B. Then one ball is drawn from Urn B and transferred to Urn
A. Let X = the number of green balls in Urn A after this process.
List the possible values for X and then find the entire probability
distribution for X.

An urn contains 25 red, 21 green, and 11 yellow balls.
Draw two balls without replacement. What is the the probability
that both balls in the sample are red?probability =
What is the probability that the number of red balls in the
sample is exactly 1 or the number of yellow balls in the sample is
exactly 1 (or both)?probability =
Draw two balls with replacement. What is the
probability that the number of red balls in the sample is...

An urn has 6 red and 4 white balls. Two balls are chosen at
random and without replacement.
Let Y be the number of red balls among those selected.
a. Find the probability function (pmf) of Y.
b. Find the moment-generating function of Y.

An urn contains 29 red, 22 green and 10 yellow balls. Draw two
balls with replacement. What is the probability
that the number of red balls in the sample is exactly 1 or the
number of yellow balls in the sample is exactly 1 (or both)?

Two balls are chosen randomly from an urn containing 6 red and 4
black balls, without replacement. Suppose that we win $2 for each
black ball selected and we lose $1 for each red ball selected. Let
X denote the amount on money we won or lost.
(a) Find the probability mass function of X, i.e., ﬁnd P(X = k) for
all possible values of k.
(b) Compute E[X].
(c) Compute Var(X)

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