Question

You have a six-sided die that you roll once. Let Ri denote the event that the roll is i. Let G j denote the event that the roll is greater than j. Let E denote the event that the roll of the die is even-numbered.

(a) What is P[RsIG1], the conditional probability that 3 is rolled given that the roll is greater than 1?
(b) What is the conditional probability that 6 is rolled given that the roll is greater than 3?
(c) What is P[GIE], the conditional probability that the roll is greater than 3 given that the roll is even?
(d) Given that the roll is greater than 3, what is the conditional probability that the roll is even?

Answer 1

a.
`P (R3 | G1)=(1)/(5)`

b.
`P (R6| G3)= (1)/(3)`

c.
`P(G3|E)=(2)/(3)`

d.
`P (E|G3)=(2)/(3)`

Explanation:

The sample space associated with the random experiment of throwing a dice is is the equiprobable space {R1, R2, R3, R4, R5, R6}. Then,

a. The conditional probability that 3 is rolled given that the roll is greater than 1?
`P (R3 | G1) = (P (R3\bigcap G1))/(P(G1)) = (1/6)/(5/6) = (1)/(5)`

b. What is the conditional probability that 6 is rolled given that the roll is greater than 3?
`P (R6| G3) = (P (R6\bigcap G3))/(P(G3)) = (1/6)/(3/6) = (1)/(3)`

c. What is P [GIE], the conditional probability that the roll is greater than 3 given that the roll is even?
`P(G3|E) = (P (G3\bigcap E))/(P(E)) = (2/6)/(3/6) = (2)/(3)`

d. Given that the roll is greater than 3, what is the conditional probability that the roll is even?
`P (E|G3) = (P (E\bigcap G3))/(P(G3)) = (2/6)/(3/6) = (2)/(3)`