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 Question from Mary, a student: I have a test and I need to know how to figure out how to find the probability of rolling 2 dice and coming up with a 7 for both rolls. Could you please explain how to find the probability? Thank you. Mary

Hi Mary,

I think you want to roll two dice twice and find the probability of rolling a 7 on both rolls of the two dice. Suppose that the two dice are different colours, one is red and the other is green. When you roll the dice you can get many possible outcomes, for example the red die might be 4 and the green die 1 or the red die might be 5 and the green die also 5. I can list all the possible outcomes using a table. I have the result on the red die along the top and the result on the green die in the left column.

1 2 3 4 5 6
1 1,1 1,2 1,3 1,4 1,5 1,6
2 2,1 2,2
3
4
5
6

Inside the table I have the result of the two die roll in the form g,r where g is the result on the green die and r is the result on the red die. I only completed part of the table, you can do the rest. There are 36 possible outcomes in this table.

If the dice are fair then the probability that you get a particular result, like 2,1 is the same as the probability that you get any other particular result. Thus the probability that you get a particular result like 2,1 is 1/36.

Usually when you roll two dice you add together the numbers on the two dice and report this sum which is an integer between 2 and 12. That is what I started in the table below.

1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4
3
4
5
6

The sum on the two dice

Complete the table above and then count how many of the outcomes is 7. Suppose this number is k. Since each of the outcomes has probability 1/36 the probability that you roll a 7 is k/36.

Now roll the two dice again. The probability that you roll a 7 this time is also k/36. Since these events are independent, that is the outcome on the second roll is not influenced by the outcome on the first roll, the probability of rolling a 7 on both rolls ia k/36 k/36.

Penny

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