# Probability with Dice.

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Two fair dice are tossed, and the face on each die is observed.

a. Use a tree diagram to find the 36 sample points contained in the same space.

b. Assign probabilities to the sample points in part a.

c. Find the probability of each of the following events:

A = {3 showing on each die}

B = {sum of two numbers showing is 7}

C = {sum of two numbers showing is even}

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A pair of fair dice is tossed. Define the following events:

A: {You will roll a 7} (i.e., the sum of the dots on the up faces of the two dice is equal to 7)

B: {At least one of the two dice shows a 4}

a. Identify the sample points in the events A, B, A upside down U B, A U B, and A^c.

b. Find P(A), P(B), P(A upside down U B), P(A U B), and P(A^c) by summing the probabilities of the appropriate sample points.

c. Find P(A U B) using the additive rule. Compare your answer to that for the same event in part b.

d. Are A and B mutually exclusive? Why?

#### Solution Preview

1

(a) and (b)

The tree and the assigned probabilities of each outcome is shown below:

(c)

1.

The probability of each die show 3 is 1/36

2.

The sum of the numbers is 7:

The possible outcomes (each with probability 1/36) are:

(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)

Hence the probability that the sum is 7 is:

3.

Sum of two numbers showing is even:

If the first die is odd, there are 3 possible outcomes that the sum is even (the second die must be 1,3, or 5)

If the first die is even, there are 3 possible outcomes that the sum is even (the second die must be 2,4, or 6)

There are 6 possible outcomes for ...

#### Solution Summary

The probability with dice are examined. The expert uses a tree diagram to find the 36 sample points contained in the same space.