Question

An urn contains three red balls, two blue balls and one yellow ball. Our experiment is to draw a ball from an urn, replace it, and draw another. Define a random variable δ: Ω → R by δ(ω) = 1 if you draw the same color twice in a row, and δ(ω) = 0 otherwise. What is the expected value of δ?

Answer #1

3 red balls, 2 blue balls and 1 yellow ball.

First we find the probabilities from drawing balls from the
urn.

Probability of drawing same color ball twice with replacement-
P(Two red balls) + P(Two blue balls) + P(Two yellow balls)

P(Two red balls) =
(first draw) x
(second draw) =
= 0.25

P(Two blue balls) =
(first draw) x
(second draw) =
= 0.11

P(Two yellow balls) =
(first draw) x
(second draw) =
= 0.028

Probability of drawing same color ball twice with replacement =
0.25 + 0.11 + 0.028 = **0.388**

Probability of drawing different ball each time = 1 - 0.625 =
**0.612**

9.84. Suppose an urn has b blue balls and r red balls. An
experiment consists of picking a ball at random from the urn, note
its color and then put it back into the urn and add one more ball
of the same color to the urn. Suppose we repeat this experiment
twice. What is the probability that we get a blue ball in the
second attempt?

An urn contains one green ball，one yellow ball，one red
ball,one white ball. I draw 4 balls with replacement. What is the
probobility that at least one color is repeated exactly twice.
(Plase show the logic behind this, thanks!)

Suppose that:
Urn U1 contains 3 blue balls and six red balls, and
Urn U2 contains 5 blue ball and 4 red balls
Suppose we draw one ball at random from each urn. If the two
balls drawn have different colors, what is the probability that the
blue ball came from urn U1?

An urn contains 9 red balls, 7 blue balls and 6 green
balls. A ball is selected and its color is noted then it is placed
back to the urn. A second ball is selected and its color is noted.
Find the probability that the color of one of the balls is red and
the color of the other ball is blue.
A. 0.2603
B. 0.2727
C. 0.4091
D. 0.3430

Question2.
An urn initially contains r red balls and b blue balls. In each
step, a ball is chosen uniformly at random, and then put back into
the urn together with a new ball of the same color. Let Ri be the
event that in step i a red ball is chosen from the urn. Show that
P(R1 ∩ R2) = P(R2 ∩ R3).

An urn contains 7 red balls, 18 blue balls and 15
green balls. A ball is selected and its color is noted and then it
is placed back to the urn. A second ball is selected and its color
is noted. Find the probability of that both balls has the same
color.
A. 0.1575
B. 0.3738
C. 0.3750
D. 0.1750

An urn contains 6 green ball, 7 blue balls and 5 yellow balls.
You are asked to draw 3 balls, one at a time (without replacement).
Find the probability that a green is pulled first, then another
green ball then a blue ball.

An urn contains 1 green ball, 1 red ball, 1 yellow ball and 1
white ball. I draw 5 balls with replacement. What is the
probability that exactly 2 balls are of the same color?

1. An experiment consists of drawing balls from an urn which
contains 2 red balls, one white ball, and one
blue ball. The balls are drawn, without replacement, until
either a blue ball has been drawn or two different
colors have been drawn. If an outcome of this experiment
consists of an ordered list of the colors of the
balls drawn, how may outcomes exist?
2. An experiment consists of repeatedly drawing a ball from an
urn which contains 3...

In an urn, there are 20 balls of four colors: red, black, yellow
and blue. For each color, there are 5 balls and they are numbered
from 1 to 5.
1) If one ball is randomly drawn from the urn, what is the
probability that the randomly selected ball is red or blue?
2) If one ball is randomly drawn from the urn, what is the
probability that the randomly selected ball is numbered 1 or
blue?

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