Question

A box contains 10 red balls, 10 white balls, and 10 blue balls. Five balls are selected at random, without replacement. Let X be the number of colors will be missing from the selection. Determine the probability mass function of X.

Answer #1

Total number of balls: 10+10+10=30

Number of ways of selecting 5 balls out of 30 is C(30,5).

At most 2 colors can be missing from the selection so X can take values 0, 1 and 2.

**When X=1:**

That is 1 color is missing. It means that sample has 2 colors. That is sample has 2 colors: red and white or red and blue or white and blue

Number of ways of selecting 5 balls from 10 red and 10 white balls:

Number of ways of selecting 5 balls out of 20 (red and white balls) is C(20,5). Number of ways of selecting 5 balls of only white colors C(10,5). Number of ways of selecting 5 balls of only red colors C(10,5). We need to find the number of ways of selecting at least one red and at least one white ball is

C(20,5) -2* C(10,5) = 15000

Likewise number of ways of selecting 5 balls from 10 red and 10 blue balls: 15000

Likewise number of ways of selecting 5 balls from 10 white and 10 blue balls: 15000

That is number of ways of selecting exactly 2 colors is: 15000 + 15000 +15000 = 45000

So,

P(X=1) = 45000 / C(30,5) = 0.3158

**When X=2:**

That is 2 colors are missing. It means that sample has 1 color. That is sample has 1 color: red or white or blue

Number of ways of selecting 5 balls from 10 red: C(10,5) = 252

Number of ways of selecting 5 balls from 10 white: C(10,5) = 252

Number of ways of selecting 5 balls from 10 blue: C(10,5) = 252

That is number of ways of selecting exactly 1 color is: 252+252+252= 756

So,

P(X=2) = 756 / C(30,5) = 0.0053

Now,

P(X=0) = 1 - P(X=1) -P(X=2) = 1 - 0.3158 - 0.0053 = 0.6789

Following table shows the pdf:

X | P(X=x) |

0 | 0.6789 |

1 | 0.3158 |

2 | 0.0053 |

A box contains "r" red balls, "w" white balls, and "b" blue
balls. Suppose that balls are drawn from the box one at a time, at
random, without replacement. What is the probability that all "r"
red balls will be obtained before any white balls are obtained?
The answer is (r!)(w!)/(r+w)! Can anyone explain why please?

A jar contains 30 red balls and 20 white balls. Twenty-five
balls are randomly selected from the jar with replacement. What is
the probability that a red ball was selected more than 20
times?
Answer: (The answer was wrong)
0.009
A jar contains 30 red balls and 20 white balls. Twenty-five
balls are randomly selected from the jar without replacement. What
is the probability that the selection contains more than 20 red
balls?
Answer: (The answer was wrong)
0.0006

A box contains 8 red and 5 white balls. 8 balls are selected at
random, without replacement. Find the probability that 3 white
balls are selected.

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.

For this problem, assume the box contains 8 blue balls, 5 red
balls, and 4 white balls, and that we choose two balls at random
from the box.
What is the probability of neither being blue given that neither
is red?

A box contains 5 blue, 10 green, and 5 red chips. We draw 4
chips at random and without replacement. If exactly one of them is
blue, what is the probability mass function of the number of green
balls drawn?

A box contains 5 blue balls, 8 red balls, 10 green balls. Ten
balls are selected from the box simultaneously. Find the expected
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have the answer but don't understand where it comes from.

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).

Refer to Example 4.40. An urn contains eight red balls, eight
white balls, and eight blue balls, and sample of five balls is
drawn at random without replacement.
Compute the probability that the sample contains at least one ball
of each color. (Round your answer to four decimal places.)

A box contains 700 blue and 300 red balls. 200 balls are
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have. (b) Find the probability mass function of X. (c) Find E[X].
(d) Find Var(X). (e) What is the probability that exactly 50 of the
balls chosen are red?

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