An unfair die is tossed. Determine which of the following assignments could be made for the probabilities of each simple outcome. If an assignment cannot be made, tell why.

Question

Question 1 part 1An unfair die is tossed. Determine which of the following assignments could be made for the probabilities of each simple outcome. If an assignment cannot be made, tell why.

(a) 𝑃(1) = 0.3, 𝑃(2) = 0.2, 𝑃(3) = −0.3, 𝑃(4) = 0.1, 𝑃(5) = 0.3, 𝑃(6) = 0.4 (b) 𝑃(1) = 0,𝑃(2) = 0.2,𝑃(3) = 0.3,𝑃(4) = 0.1,𝑃(5) = 0.5,𝑃(6) = 0.1

(c) 𝑃(1) = 0.1, 𝑃(2) = 0.2, 𝑃(3) = 0.3, 𝑃(4) = 0.1, 𝑃(5) = 0.1, 𝑃(6) = 0.2



Question 1 part 2 If we roll a fair die twice,

(a) What is the sample space?

(b) what is the probability that the total sum is 5? (Write the event as well)

(c) what is the probability that the total sum is less than 5? (Write the event as well) (d) What is the probability of obtaining two even numbers? (Write the event as well) (e) what is the probability that the total sum is more than 12? (Write the event as well)

Answer 1

(a) 𝑃(1) = 0.3, 𝑃(2) = 0.2, 𝑃(3) = −0.3, 𝑃(4) = 0.1, 𝑃(5) = 0.3, 𝑃(6) = 0.4

Negative probabilities are impossible! This assignment cannot be made.

(b) 𝑃(1) = 0,𝑃(2) = 0.2,𝑃(3) = 0.3,𝑃(4) = 0.1,𝑃(5) = 0.5,𝑃(6) = 0.1

These probabilities add to more than 1. They are supposed to represent mutually exclusive but exhaustive set of outcomes, so the probabilities should add to 1. This assignment cannot be made.

(c) 𝑃(1) = 0.1, 𝑃(2) = 0.2, 𝑃(3) = 0.3, 𝑃(4) = 0.1, 𝑃(5) = 0.1, 𝑃(6) = 0.2

Looks good. All probabilities are between 0 and 1, and they add to 1.

Question 1 part 2 If we roll a fair die twice,

(a) What is the sample space?

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

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

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

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

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

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

(b) what is the probability that the total sum is 5? (Write the event as well)

Let X represent the sum of rolls.

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

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

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

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

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

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

P(X=5) = 4/36 = 1/9

(c) what is the probability that the total sum is less than 5? (Write the event as well) 

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

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

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

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

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

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

P(X<5) = 6/36 = 1/6

(d) What is the probability of obtaining two even numbers? (Write the event as well)

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

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

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

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

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

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

P(both even) = 9/36 = 1/4

Also, P(both even) = P(1st even)*P(2nd even) = 1/2 * 1/2 = 1/4

 (e) what is the probability that the total sum is more than 12? (Write the event as well)

The maximum sum is 12.

P(X>12) = 0