I REALLY NEED HELP WITH THIS COMPOUND PROBABILITY QUESTION?????

Question

An ordinary (fair) die is a cube with the numbers 1

 through 6

 on the sides (represented by painted spots). Imagine that such a die is rolled twice in succession and that the face values of the two rolls are added together. This sum is recorded as the outcome of a single trial of a random experiment.Compute the probability of each of the following events.

Event A

: The sum is greater than 8

.

Event B

: The sum is an odd number.

Answer 1

When you roll two six sided dice, there are six possibilities for each of the dice. To find the total number of possibilities you multiply each of the individual possibilities, so 6 x 6 = 36 is how many ways the dice can end up. Another way to think of this is you can have 1 through 6 on the first die, and 1 through 6 on the second die for 36 combinations. The dice are fair, so each number, and each combination has the same probability of occurring.

For event A, a sum greater than 8, you need to know how many ways you can get a 9, a 10, an 11, or a 12. All of those are greater than 8, so once you know thjose separate probabilities, you add them together to get the overall probability.

For a 9, You can have a 6 and a 3, or a 5 and a 4, and this can happen on either of the dice, so the are 4 ways to get a 9. P(9) = 4/36

For a 10, You can have a 6 and a 4 on either of the dice, or you can have two 5's. There are 3 possibilites, so P(10) = 3/36.

For an 11, you need a 5 on one die and a six on the other, so there are two ways to get an 11, and P911) = 2/36

The only way to get a 12 is having a six on each of the dice, so P(12) = 1/36.

Now add up P(9), P(10), P(11), and P(12) to get the probability of a sum greater than 8:

4/36 + 3/36 + 2/36 + 1/36 = 10/36 = 5/18,

So, P(sum>8) = 5/18.

For event B, a sum that is odd, you need an odd number on one die, and an even number on the other die

When you roll the first die, there is a 1/2 probability of an odd number. Then when yuou roll the second die you would need an even number for the sum to be odd. There is a 1/2 probability for that to occur. So if you roll an odd number on the first die there is a 1/2 x 1/2 = 1/4 probability of the sum being odd.

If, however you rolled an even number on the first die, there would be a 1/2 probability of that to happen, and now you would need an odd number on the second die, which would also have a 1/2 probability. Multiply the two probabillities like you did before, and that probability would be 1/2 x 1/2 = 1/4.

So there are two ways to get an odd number, and both have probabilities of 1/4. The are mutually exclusive (when cannot happen at the same time as the other) so you can add the probabilities to get the overall probability of rolling an odd total

P(odd) = 1/4 + 1/4 = 1/2

Hope that helps