Question

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).

Answer #1

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).

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.

An urn contains 15 red balls, 10 black balls and 25 white balls.
10 balls are randomly picked. We define
to random variables N as the number of black balls picked and to B
as the number of white balls picked.
a)Calculate the joint mass function of N and B.
b)Calculate the joint distribution function of N and B.

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.

An urn contains 7 balls: 2 white, 3 green, and 2 red.
We draw 3 balls without replacement. Find the probability that
we don’t see all three colors.Probability =
We draw 3 balls with replacement. Find the probability that we
don’t see all three colors.Probability =

Urn 1 contains 4 red balls and 3 black balls. Urn 2 contains 1
red ball and 3 black balls. Urn 3 contains 4 red balls and 2
black balls. If an urn is selected at random and a ball is
drawn, find the probability that it will be red.
Enter your answer as a fraction in simplest form or a decimal
rounded to 3 decimal places.
P(red)=

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 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.

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)

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.

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