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Probability of Two Dice Summing to 5

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?

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 

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   

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   
Associated Topics:
High School Probability
Middle School Probability

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