Question

I. Consider the random experiment of rolling a pair of dice. Note: Write ALL probabilities as reduced fractions or whole numbers (no decimals).

1-1	2-1	3-1	4-1	5-1	6-1
1-2	2-2	3-2	4-2	5-2	6-2
1-3	2-3	3-3	4-3	5-3	6-3
1-4	2-4	3-4	4-4	5-4	6-4
1-5	2-5	3-5	4-5	5-5	6-5
1-6	2-6	3-6	4-6	5-6	6-6

2) How many outcomes does the sample space contain? _____36________

3)Draw a circle (or shape) around each of the following events (like you would circle a word in a word search puzzle). Label each event in the sample space with the corresponding letter.

A: Roll a sum of 3.

B: Roll a sum of 6.

C: Roll a sum of at least 9.

D: Roll doubles.

E: Roll snake eyes (two 1’s). F: The first die is a 2.

3) Two events are mutually exclusive if they have no outcomes in common, so they cannot both occur at the same time.

Are C and F mutually exclusive? ___________

Using the sample space method (not a special rule), find the probability of rolling a sum of at least 9 and rolling a 2 on the first die on the same roll. P(C and F) = __________

Using the sample space method (not a special rule), find the probability of rolling a sum of at least 9 or rolling a 2 on the first die on the same roll.

P(C or F) = __________

4) Special case of Addition Rule: If A and B are mutually exclusive events, then

P(A or B) = P(A) + P(B)

Use this rule and your answers from page 1 to verify your last answer in #6:

P(C or F) = P(C) + P(F) = ________ + ________ = _________

5) Are D and F mutually exclusive? __________

Using the sample space method, P(D or F) = _________

6) Using the sample space method, find the probability of rolling doubles and rolling a “2” on the first die.

P (D and F) = _______

7) General case of Addition Rule: P(A or B) = P(A) + P(B) – P(A and B)

Use this rule and your answers from page 1 and #9 to verify your last answer in #8:

P(D or F) = P(D) + P(F) – P(D and F) = ________ + ________ − ________ = _________

8) Two events are independent if the occurrence of one does not influence the probability of the other occurring. In other words, A and B are independent if P(A|B) = P(A) or if P(B|A) = P(B).

Compare P(D|C) to P(D), using your answers from page 1: P(D|C) = ________ P(D) = ________ Are D and C independent? _________ because _______________________________

When a gambler rolls at least 9, is she more or less likely to roll doubles than usual? ___________ Compare P(D|F) to P(D), using your answers from page 1: P(D|F) = ________ P(D) = ________

Are D and F independent? __________ because ______________________________

9) Special case of Multiplication Rule: If A and B are independent, then P(A and B) = P(A) · P(B).

Use this rule and your answers from page 1 to verify your answer to #9: P(D and F) = P(D) • P(F) = ________ · ________ = ________ .

10) Find the probability of rolling a sum of at least 9 and getting doubles, using the sample space method.

P(C and D) = ___________ .

11) General case of Multiplication Rule: P(A and B) = P(A) · P(B|A).

Use this rule and your answers from page 1 to verify your answer to #13: P(C and D) = P(C) • P(D|C) = ________ · ________ = ________ .

Answer 1

3.

A.

B.

C.

D.

E.

F.

3.

We observe that the events C and F have no outcome in common. So these two events are mutually exclusive.

[As being mutually exclusive, no outcomes in common]

We observe that there are 16 outcomes of which sum is either greater than 9 (circled in blue) or first die is a 2 (circled in red).

4.

5..

We observe that the events D and F have a outcome (2,2) in common. So these two events are not mutually exclusive.

We observe that there are 11 outcomes of which sum is double is rolled (circled in blue) or first die is a 2 (circled in red).

6.

We observe that there is a outcome (2,2) which sum is double as well as the first die is a 2

.

7.

8.

So, D and C are not independent because P(D|C) differs from P(D).

So, D and F are independent because P(D|F) is same as from P(D).

9.

10.

There are 2 outcomes whose sum are at least 9 and which are doubles.

11.