Probability of a pair of dice

Question

Consider rolling a pair of fair dice a single time. For a given roll, consider the value shown on each of the dice. For each of the following pairs of events, tell whether they are disjoint, independent, or neither, and find the probabilities for both events occurring.

(a) Event A: The value of the first dice is 3.

Event B: The value of the second dice is greater than 4.

(b) Event A: Either of the first two dice are even.

Event B: Either of the first two dice are divisible by 3.

(c) Event A: The value of the first dice is less than 2.

Event B: The value of the first dice is greater than 4.

(d) Event A: The value of the second dice is greater than 3.

Event B: The value of the second dice is greater than 5.

Answer 1

Let's take this in parts:

DISJOINT: To answer whether they are disjoint, you need only determine if both events can happen simultaneously or not. For example, for the first part, we can clearly see that both of these events could occur at the same time (e.g. rolling a 3 on the first die and a 5 on the second die would satisfy both events at the same time). Therefore, the events are not disjoint. Just do that same "test" for all four parts (you should get No, No, Yes, No)

INDEPENDENT: To test for independence, it is much easier to use the formula for independent events (P(A)•P(B) = P(A and B)) than to try to reason this out conceptually. Basically, if the formula is true for the given events, they are independent. If it is not true, they aren't, simple as that! Let's use the first example again:

P(A) = 1/6

P(B) = 2/6 or 1/3 (could roll a 5 or a 6 on that die)

P(A and B) has two possible outcomes: rolling a 3 then a 5 or rolling a 3 then a 6. Therefore, P(A and B) = 2/36 or 1/18. So now, run the test using the calculated probabilities:

1/6 • 1/3 = 1/18 is TRUE, so the events are independent.

The nice thing is that by answering the question about independence in this way, you will already have calculated the probabilities of both events occurring.

Answer 2

P(A) = 1/6, P(B) =2/6 =1/3

P(A or B) + P(A)+P(B) - P(A&B)= 1/6 + 1/3 - (1/6)(1/3) = 1/2- 1/18= 8/18=4/9 = about44%

independent, disjoint event 50% probability of either occuring

Probability both A and B occur = 44%

P(either even) = P(A)= P(2,4or6) = 1/2

P(divisible by 3) = P(B) = 2/6 = 1/3

P(A or B) + P(A)+P(B)- P(A&B) = 1/2+ 1/3 - 1/6 =2/3= about 67% chance of either happening

neither disjoint nor independent

Probability of both happening =1/6 = about 18%

A = 1, B= 5 or 6

P(A) = 1/6, P(B) = 1/3

P(A&B) = 1/6 x 1/3 = 1/18

disjoint, independent

A=4,5or 6, B= 6

P(A)= 1/2, P(B) = 1/6

P(A&B) = 1/2 x 1/6 = 1/12

not disjoint, independent

Answer 3

(A) the outcomes are disjoint and independent.

(B) the outcomes are neither disjoint nor independent.

(C) the outcomes are disjoint and dependent (that is, not independent).

(D) the outcomes are neither disjoint nor independent.