# What Are the Probability Mass Functions for Randomly Selected Marbles?

• Jimerd
In summary, the probability of getting a specific outcome in a sample of 12 marbles is calculated by multiplying the probability of each outcome by the number of times it occurs in a sample.
Jimerd
A box contains 12 marbles. Six of the marbles are red, four are green, and two are yellow. Suppose that you choose three marbles at random. let x be the number of red marbles in the sample, y be the number of green marbles, and z be the number of yellow marbles.

a. Give the precise formulas for the probability mass functions of the three random variables, f(x), f(y), and f(z).

b. Suppose you win $1 for each green marble in your sample and lose$1 for each red marble (yellow has no financial consequences). Let w be your net winnings or losses. Construct the P.M.F of w as a table. (obviously the table part cannot be shown in this answer)

c. Find the expected value of w and explain its significance.

d. Find the standard deviation of w.

Any help is greatly appreciated! If anyone can point me in the right direction, that'll be great!

Look up the hypergeometric distribution.

Yes, thank you. I put it into the formula and it gave me the correct answer :)

Jimerd said:
Yes, thank you. I put it into the formula and it gave me the correct answer :)

More to the point: do you know *why* the appropriate distribution is the hypergeometric? BTW: can you see why using the simple hypergeometric alone will not help you do part (d)?
(Hint: you need a covariance between two random variables, so you need a bivariate distribution.)

RGV

This is what I got for part (a)

F(x)= 3Ck * 0.5^3
F(y)=3Ck * 0.667^3
F(z)=3Ck * 0.167^3

Now for part (b) I have some problems, I don't know if I got all the possible outcomes

RRR
RRG
RGR
GRR
GGG
GGR
GRG
RGG
RGY
RYG
GYR
GRY
YRG
YGR
YYR
YYG
YRY
RYY
YGY
GYY
YYY

I got the probability of each and how much cash you would have.

Jimerd said:
This is what I got for part (a)

F(x)= 3Ck * 0.5^3
F(y)=3Ck * 0.667^3
F(z)=3Ck * 0.167^3

Now for part (b) I have some problems, I don't know if I got all the possible outcomes

RRR
RRG
RGR
GRR
GGG
GGR
GRG
RGG
RGY
RYG
GYR
GRY
YRG
YGR
YYR
YYG
YRY
RYY
YGY
GYY
YYY

I got the probability of each and how much cash you would have.

Your formulas for F(x), F(y) and F(z) are all incorrect. You have NOT used the hypergoemetric distributions here.

Let's look at choosing two reds in 3 draws (starting with a box of 6 Red, 4 green, 2 yellow) Suppose we label the outcomes as R (red) and N (not red). What is the probability of the specific outcome RRN? Look at it in pieces: what is the probability of R1: the first draw is red? There are 6 reds in 12 balls, so P{R1} = 6/12. Now you have 11 balls left and 5 are red, so what now is the probability of the second draw = red (R2)? Remember, we have already observed R1, so we are really asking for P{R2|R1} = 5/11. Now we have 10 balls left, of which 4 are red and 6 are non-red. So, P{N3|R1,R2} = 6/10. Altogether, P{R1 R2 N3} = P{RRN} = (6/12)(5/11)(6/10).

You can work out similar probabilities for RNR and NRR, and add them up. That will give you P{2 red}.

Once you have done this a few times you will realize there are slick shortcut formulas allowing you to write down the answer quickly, but first you need to realize what is involved.

RGV

## 1. What is a Probability Mass Function (PMF)?

A Probability Mass Function (PMF) is a mathematical concept used in statistics and probability theory to describe the probability of a discrete random variable taking on a specific value. It maps each possible value of the random variable to its corresponding probability.

## 2. How is a PMF different from a Probability Density Function (PDF)?

The main difference between a PMF and a PDF is that a PMF is used for discrete random variables, while a PDF is used for continuous random variables. This means that a PMF gives the probability of a specific value, while a PDF gives the probability of a range of values.

## 3. How can a PMF be represented graphically?

A PMF can be represented graphically using a bar chart, with the values of the random variable on the x-axis and the corresponding probabilities on the y-axis. The height of each bar represents the probability of that particular value occurring.

## 4. What is the relationship between a PMF and a Cumulative Distribution Function (CDF)?

A PMF and a CDF are closely related, as the CDF is the cumulative sum of the PMF. This means that the CDF gives the probability of a random variable being less than or equal to a specific value, whereas the PMF gives the probability of a specific value occurring.

## 5. How can a PMF be used in practical applications?

A PMF can be used in various practical applications, such as in market research, insurance risk assessment, and quality control. It helps in understanding and predicting the likelihood of certain events or outcomes occurring, which can aid in decision-making and planning.

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