Question

Three fair dice (black, white and red) are tossed simultaneously and the value recorded as an ordered triple y = (yb, yw, yr), which is a point in S.

Let A1, A2, A3 be the events A1 = {(yb, yw, yr) : yb = yw} A2 = {(yb, yw, yr) : yb = yr} A3 = {(yb, yw, yr) : yw = yr}

For the set-up described in the preceding question, let 1 ≤ t ≤ 18 be a positive integer, and let Bt ⊂ S be the event that the total is t: Bt = {(yb, yw, yr) ∈ S : yb + yw + yr = t}.

For example, B4 is the event that the sum is 4.

(i) The events B1 and B2 do not exist. (True/False?)

(ii) The events B1 and B2 exist and are equal. (True/False?)

(ii)’ The events B1 and B2 are equal but they do not exist. (True/False?)

(iii) The events B1 and B4 are disjoint and independent. (True/False?)

(iv) The events B4 and B6 are disjoint. (True/False?)

(v) The events B4 and B6 are independent. (True/False?)

(vi) List the points in B4.

(vii) List the points in B5.

(viii) List the points in A1 ∩ B5.

(ix) Compute P(A1 | B5).

(x) Compute P(B5 | A1).

Answer 1

i) The minimum value of the sum is . Therefore, events B1 and B2 do not exist. TRUE.

ii) The statemnet is FALSE because if (i).

ii) (nullset). TRUE.

iii) Since the sums (1 and 4) are different, are disjoint. Since , they are also independent.

TRUE.

iv) Since the sums (4 and 6) are different, are disjoint. But not indepenednt. TRUE.

v) are nonzero. But . S they are not indepenednt. FALSE