Suppose you roll 2 dice 1 red and 1 blue Let Rn be the event

Suppose you roll 2 dice (1 red and 1 blue). Let Rn be the event that the red die comes up on number n and likewise Bm be the event that the blue die comes up on number m. Calculate the following:

(a) Pr(R1&B1)

(b) Pr(R1 or B1)

(c) P r((R1 or R2)&(B1 or B2))

(d) Pr(not (R1 or B1))

(e) Pr((R1&R2) or (B1&B2))

(f) Pr(R1 \"given that\" R1 or R2 or R3)

Solution

a) Pr(R1&B1)

P(R1) = 1/6 , P(B1) = 1/6 , so Pr(R1&B1) = P(R1)*P(B1) = 1/6 * 1/6 = 1/36

(b) Pr(R1 or B1) = 1/6 + 1/6 =2/6 = 1/3

(c) P r((R1 or R2)&(B1 or B2))

P(R2) = 1/6 and P(B2) = 1/6 , so P r((R1 or R2)&(B1 or B2)) = (1/6+1/6) * (1/6+1/6) = (1/3)*(1/3) = 1/9

(d) Pr(not (R1 or B1)) = 1- ( 1/6+1/6) = 1-(1/3) = 2/3

(e) Pr((R1&R2) or (B1&B2)) = (1/6 * 1/6)+ (1/6 * 1/6) = 2/36 = 1/18

(f) Pr(R1 \"given that\" R1 or R2 or R3) = P(R1 and(R1 or R2 or R3)) / P(R1 or R2 or R3)

  P(R1 or R2 or R3) = 3/6 = 1/2 P(R1 and(R1 or R2 or R3)) = 1/6 * 1/2 = 1/12

so Pr(R1 \"given that\" R1 or R2 or R3) = (1/12) / (1/2) = 2/12 = 1/6