Question

Box 1 contains 4 red balls, 5 green balls and 1 yellow ball.

Box 2 contains 3 red balls, 5 green balls and 2 yellow balls.

Box 3 contains 2 red balls, 5 green balls and 3 yellow balls.

Box 4 contains 1 red ball, 5 green balls and 4 yellow balls.

Which of the following variables have a binomial
distribution?

(I) Randomly select three balls from Box 1 with replacement. X =
number of red balls selected

(II) Randomly select one ball from each of the four boxes. X = number of yellow balls selected

(III) Randomly select three balls from Box 1 without replacement. X = number of red balls selected

(IV) Randomly select one ball from each of the four boxes. X = number of green balls selected

Answer #1

(I)

Since ball are drawn with replacement so probability of getting a red ball is p = 4/10 each time.

It is binomial distrbution with parameters n=3 and p= 0.40.

(II)

Since probability of getting yellow ball from each box is different so it is not binomial.

(iii)

Since ball are drawn with replacement so it is not binomial. In this case probability of getting a red ball is not same each time.

(iv)

The probability of getting green ball from each is 5/10 = 0.20

So here X has binomial distribution with parameter n = 1+1+1+1= 4 and p=0.20.

A box contains 4 red balls, 3 yellow balls, and 3 green balls.
You will reach into the box and blindly select a ball, take it out,
and then place it to one side. You will then repeat the experiment,
without putting the first ball back. Calculate the probability that
the two balls you selected include a yellow one and a green
one.

A box contains one yellow, two red, and three green balls. Two
balls are randomly chosen without replacement. Define the following
events:
A:{ One of the balls is yellow }
B:{ At least one ball is red }
C:{ Both balls are green }
D:{ Both balls are of the same color }
Find the following conditional probabilities:
P(B\Ac)=
P(D\B)=

A
basket contains 5 Red balls, 1 Blue ball and 1 Green ball. Three
balls were selected randomly without replacement. Find the
probability that the three selected balls contain at least two red
balls.

A box contains one yellow, two red, and three green balls. Two
balls are randomly chosen without replacement. Define the following
events:
A: \{ One of the balls is yellow \}
B: \{ At least one ball is red \}
C: \{ Both balls are green \}
D: \{ Both balls are of the same color \}
Find the following conditional probabilities: (a) P(B|D^c)
(b) P(D|C)
(c) P(A|B)

Box I contains 7 red and 3 black balls; Box II contains 4 red
and 5 black balls. After a randomly selected ball is transferred
from Box I to Box II, 2 balls are drawn from Box II without
replacement. Given that the two balls are red, what is the
probability a black ball was transferred?

2. Box A contains 10 red balls, and 15 green balls. Box B
contains 12 red balls and 17 balls
green. A ball is taken randomly from box A and then returned to box
B.
From box B a random ball is drawn.
a) Determine the chance that two green balls are taken.
b) Determine the chance that 1 red ball is drawn and 1 green ball
is taken

1. You have three boxes labelled Box #1, Box #2, and
Box #3. Initially each box contains 4 red balls
and 4 green balls. One ball is randomly selected from
Box #1 and placed in Box #2 Then one ball is randomly selected from
Box #2 and placed in Box #3. Then one ball is randomly selected
from Box #3 and placed in Box #1. At the conclusion of
this process, what is the probability that that each box has the
same number of...

There are six red balls and three green balls in a box. If we
randomly select 3 balls from the box with replacement, and let X be
number of green balls selected. So, X ~ Binomial [n=3, p=1/3] Use R
to find the following probabilities and answers. A) How likely do
we observe exactly one green ball? B) Find P[X<=2]. C) Find the
second Decile (the 20th percentile). D) Generating 30 random
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randomly select 3 balls from the box with replacement, and let X be
number of green balls selected. So, X ~ Binomial [n=3, p=1/3] Use R
to find the following probabilities and answers. A) How likely do
we observe exactly one green ball? B) Find P[X<=2]. C) Find the
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Exercise 2 A box contains 3 white balls, 4 red balls and 5 black
balls. A ball is picked, its color recorded and returned to the
box(with replacement). Another ball is then selected and its color
recorded.
1. Find the probability that 2 black balls are selected.
2. Find the probability that 2 balls of the same color are
selected.
Now 4 balls are picked with replacement
3.Find the probability no red balls are selected.
4.Find the probability that the...

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