Binomial Theorem > How to find the mean of the probability distribution

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## Mean of a probability distribution

Watch the video or read the steps below:

## How to find the mean of the probability distribution : Overview

Finding the

**mean**of a

**probability distribution**is easy in probability and statistics — if you know how. This how to will guide you through a few simple steps necessary to find the mean of the probability distribution or binomial distribution. You’ll often find these types of questions in textbook chapters on binomial probability distribution. The binomial distribution is just a simple trial where there are two outcomes: success or failure. For example, if you are counting how many times you draw an Ace from a deck of cards, you could assign “Success” to “Drawing an Ace” and “Failure” to drawing any other card. You can find the mean of the probability distribution by creating a probability table.

## How to find the mean of the probability distribution: Steps

**Sample question**: “A grocery store has determined that in crates of tomatoes, 95% carry no rotten tomatoes, 2% carry one rotten tomato, 2% carry two rotten tomatoes, and 1% carry three rotten tomatoes. **Find the mean** number of rotten tomatoes in the crates.”

**Step 1:***Convert all the percentages to decimal probabilities*. For example:

95% = .95

2% = .02

2% = .02

1% = .01**Step 2:***Construct a probability distribution table*. (If you don’t know how to do this, see**how to construct a probability distribution**).)

**Step 3:***Multiply the values in each column. (In other words, multiply each value of X by each probability P(X).)*

Referring to our probability distribution table:

0 × .95 =**0**

1 × .02 =**.02**

2 × .02 =**.04**

3 × .01 =**.03****Step 4:***Add the results from step 3 together*.

0 + .02 + .04 + .03 =**.09**is the mean.

You’re done finding the mean for a **probability distribution!**

## Mean of Binomial Distribution

A binomial distribution represents the results from a simple experiment where there is “success” or “failure.” For example, if you are polling voters to see who is voting Democrat, the voters that say they will vote Democrat is a “success” and anything else is a failure. One of the simplest binomial experiments you can perform is a coin toss, where “heads” could equal “success” and “tails” could equal “failure.”

The **mean of binomial distribution** is much like the mean of anything else. It answers the question “If you perform this experiment many times, what’s the likely (the average) result?.

## Formula for Mean of Binomial Distribution

The formula for the **mean of binomial distribution** is:

**μ = n *p**

Where “n” is the number of trials and “p” is the probability of success.

For example: if you tossed a coin 10 times to see how many heads come up, your probability is .5 (i.e. you have a 50 percent chance of getting a heads and 50 percent chance of a tails) and “n” is how many trials — 10. Therefore, the mean of this particular binomial distribution is:

10 * .5 = 5.

This makes sense: if you toss a coin ten times you would expect heads to show up on average, 5 times.

### Mean for a Binomial Distribution on the TI-83

**Sample problem**: Find the mean for a binomial distribution with n = 5 and p = 0.12.

Again, the TI 83 doesn’t have a function for this. But if you know the formula (n*p), it’s pretty easy to enter it on the home screen.

**Step 1:** Multiply n by p.

5 * .12 ENTER

**=.6**

Hey, that was easy!

### Something to think about:

You may be wondering *why* it was so easy to calculate the mean. After all being asked to “calculate the mean for a binomial distribution” *sounds* scary. If you think about what a mean (or average) is, then you’ll see why it was so easy. In the sample question, n = 5 and p = 0.12. What is “n”? That’s the number of items. So imagine a list of 5 items with a certain score:

1 = 0.12

2 = 0.12

3 = 0.12

4 = 0.12

5 = 0.12

If you were asked to find the average score for those five items, you wouldn’t even have to do the math: it’s just 0.12, right? Finding the mean for a binomial distribution is just a little different: you add up all of the probabilities (0.12 + 0.12 + 0.12 + 0.12 + 0.12). Or a faster way, just multiply n by p.

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I am helping my son with his homework and I want to make sure that my assumptions are correct before I help him with this question. I have modified the question a bit from his homework to simplify.

I think his question is similar to your example above:

If you have a spinner with 4 zones and you win a certain number of coins in each of the zones (number of coins listed under zone below). The probably of landing in each of the zones is unequal (given below).

The question is:

What is the mean number of coins you expect to win in a single spin in the game?

Zone Probability %

0 17

1 40

3 10

25 2

so

0*.17=0

1*.40=.4

3*.10=.3

25*.02=.5

0+.4+.3+.5=1.2 coins is the mean you can expect in a single spin

Hi, Kristen,

Can you post your question on the discussion board? Unfortunately, I don’t have the time to answer math questions here.

Thanks,

Stephanie

For a binomial distribution, it’s much faster to multiply the number of trials (n) by the probability of success (p). So, the mean for a binomial distribution is np. I couldn’t find that method on your site. So for example, if a basketball player has a 90% chance of making any given free throw attempt and we assume all the binomial conditions are met, then during a practice session where the player makes 30 attempts at a free throw, the average number of shots actually made successfully would be (30)(.9)=27.

Steve,

Thanks for your note. Step 3 does multiply p*n (technically), when you multiply the probability by the rotten tomatoes. However, this how to is for the mean (so you would be looking at the average several basketball players with several attempts each at free throws).

Stephanie

A building venice is designed by taking 20 years return period water level.

a) Calculate the probability of water level exceeding the design value less than 2 times in 20 years with Binomial distrubition.

b)Calculate the probability of water level exceeding the design value more than 2 times in 60 years with Poisson distrubition.

Were you given any probabilities for the water level exceeding the design value? Without this info, the question is impossible to answer.

Assume a banks 95% value at risk model is perfectly accurate,If daily losses are independent, what is the probability that the number of daily losses exceeds the Var on exactly 5 days out of the previous 100 trading days.

This helps so much! Thanks!