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### Probability of Two Dice Summing to 5

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Date: 09/22/2001 at 20:41:31
From: Yuxiao
Subject: Probabilities

I don't really understand how to do probabilities. For example:

If a person rolls two dice, what is the probability of getting a five
as the sum of the two dice?

Can you explain it step by step?
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Date: 09/23/2001 at 07:33:40
From: Doctor Mitteldorf
Subject: Re: Probabilities

Dear Yuxiao,

It takes a lot of getting used to. The only way to get a feeling that
you really understand probabilities is to do lots and lots of
examples.

One important principle is multiplication. You're familiar with
multiplying numbers like 7*3 = 21, where the numbers get bigger as you
multiply them. But if the numbers are fractions less than 1, then
multiplying them together makes the result smaller.

All probabilities are less than or equal to 1, so multiplying them
together makes a smaller number. If the probability of one thing
happening is x and the probability of another thing happening is y,
you can multiply x times y to get a smaller number that is the
probability of both things happening.

Let's apply this to the two dice. You know that the probability of
getting a 1 on the first die is 1/6. The probability of getting a 4 on
the second die is also 1/6. So multiply these two together and you
find that the probability of getting BOTH a 1 on the first die AND a
4 on the second die is 1/36.

That's one of the ways you can get a 5 with two dice. So 1/36 is part
of the probability of rolling a 5, but not all of it. Can you list the
other ways?

First die      Second die
1               4
2               3
3               2
4               1

We've listed four ways to get a five, and that's all there are. Each
of these combinations has a probability of 1/36 of happening; so the
total probability of rolling a 5 is 4/36, which is 1/9.

A good next step for you would be to make a chart of all the results
1 through 12 and calculate the probabilities for each in the way I
just did for 5. You can check your chart when you're finished by
adding up the probabilities for all 12 numbers: The probabilities
should add up to 1. That's because one of these numbers HAS TO come
up, so the probability of getting any number 1 through 12 is 1.

- Doctor Mitteldorf, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 09/23/2001 at 07:47:13
From: Doctor Anthony
Subject: Re: probabilities

In your example, the probability is the ratio

Number of ways we can get a total of 5
--------------------------------------
Total number of possible outcomes

From here you simply count the ways we can get 5.

1 + 4   probability of this is (1/6)(1/6) = 1/36
2 + 3        "           "                = 1/36
3 + 2        "           "                = 1/36
4 + 1        "           "                = 1/36
--------------
Total probability = 4/36 = 1/9

Alternatively, there are 4 ways we can get a total of 5 and there are
36 possible outcomes when you roll two dice.

4       1
Required probability =  ----- =  ---
36       9

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Probability
Middle School Probability

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