Question

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)

Answer #1

a) below is probability mass function of X :

P(X=-2)=P(both red balls)=(6/10)*(5/9)=1/3

P(X=1)=P(1red and 1 black ball)=2*(6/10)*(4/9)=8/15

P(X=4)=P(both black balls)=(4/10)*(3/9)=2/15

b)

x | P(x) | xP(x) |
x^{2}P(x) |

-2 | 1/3 | -0.667 | 1.333 |

1 | 8/15 | 0.533 | 0.533 |

4 | 2/15 | 0.533 | 2.133 |

total | 0.400 | 4.000 | |

E(x) =μ= | ΣxP(x) = | 0.4000 | |

E(x^{2}) = |
Σx^{2}P(x) = |
4.0000 | |

Var(x)=σ^{2} = |
E(x^{2})-(E(x))^{2}= |
3.840 |

E(X)=0.400

c)Var(X)= 3.840

Three balls are randomly chosen from an urn containing 3 white,
4 red and 5 black balls. Suppose one will win $1 for each white
ball selected, lose 1$ for each red ball selected and receive
nothing for each black ball selected. Let Random Variable X denote
the total winnings from the experiment. Find E(X).

Two balls are chosen randomly from an urn containing 5 black and
5 white balls. Suppose that we win $1 for each black ball selected
and we lose $1 for each white ball selected. Denote our winnings by
a random variable X.
(a) (4 points) Provide the probability distribution of X.
(b) (2 points) Using the result in (a), what is the probability
that 0 ≤ X ≤ 2?

Two balls are chosen randomly from an urn containing 9 yellow, 5
blue, and 3 magenta balls. Suppose that we win $3 for each blue
ball selected, we lose $2 for each yellow ball selected and we win
$0 for each magenta ball selected. Let X denote our winnings. What
are the possible values of X, what are the probabilities associated
with each value (i.e., find the probability mass function of X),
and what is the expectation value of X,E[X]?

Three balls are randomly chosen from an urn containing 3 white,
3 red, and 5 blackballs. Suppose that we win $1 for each white ball
selected, and lose $1 for each red ball selected. If X denotes our
total winnings from the experiment, then:
a) What values can X take?
b) What is the PMF of X?
c) Show that this is a valid PMF.

Two balls are chosen at random from a closed box containing 9
white, 4 black and 2 orange balls. If you win $2 for each black
ball selected but lose $1 for each white ball selected, and letting
X stand for the amount of money won or lost, what are the possible
values of X and what are the probabilities associated with each
X?
What is the expected value of your winnings? (Note: round to the
nearest dollar)
A. Lose...

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.

From an urn containing 9 red balls and 6 green balls,
4 balls are taken without replacement. Determine the probability
that all 4 ball are green
Give the probability if the same experiment is
preformed with replacement and the same outcome is obtained

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 has 12 balls. 8 are white balls and 4 are black
balls.
If we draw a sample of 3 balls (i.e., picking without
replacement) and given that the first two balls selected were a
black ball and a white ball, what is the conditional probability of
the third ball drawn being white?

There are 8 black balls and 7 red balls in an urn. If 4 balls
are drawn without replacement, what is the probability that no more
than 1 black ball is drawn? Express your answer as a fraction or a
decimal number rounded to four decimal places.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 44 seconds ago

asked 9 minutes ago

asked 27 minutes ago

asked 37 minutes ago

asked 56 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago