Question

John and Mary each have a balanced die. They each roll their die repeatedly; once someone rolls a ‘6’, he/she stops
rolling. Let Y1 = the number of times John rolls and Y2= the number of times Mary rolls.
(a) Find P(Y1 + Y2 =5).
(b) Find P(Y1 = Y2).

Answer 1

(a)

Y1 ~ Geom(p = 1/6)

Y2 ~ Geom(p = 1/6)

and Y1 and Y2 are independent.

P(Y1 = k) = (1 - 1/6)^k-1 (1/6) = (1/6) * (5/6)^k-1 for k = 1, 2, .....

Similarly,  P(Y2 = k) = (1/6) * (5/6)^k-1 for k = 1, 2, .....

P(Y1 + Y2 = 5) = P(Y1 = 1, Y2 = 4) + P(Y1 = 2, Y2 = 3) + P(Y1 = 3, Y2 = 2) + P(Y1 = 4, Y2 = 1)

= P(Y1 = 1) P(Y2 = 4) + P(Y1 = 2) P(Y2 = 3) + P(Y1 = 3) P(Y2 = 2) + P(Y1 = 4) P(Y2 = 1)

= (1/6) * (5/6)^1-1 * (1/6) (5/6)^4-1 +  (1/6) * (5/6)^2-1 * (1/6) (5/6)^3-1 +  (1/6) * (5/6)^3-1 * (1/6) (5/6)^2-1 ​​​​​​​+  (1/6) * (5/6)^4-1 * (1/6) (5/6)^1-1 ​​​​​​​

= (1/6)² * (5/6)³ +  (1/6)² * (5/6)³ ​​​​​​​+  (1/6)² * (5/6)³ ​​​​​​​+  (1/6)² * (5/6)³ ​​​​​​​

= 4 * (1/6)² * (5/6)³ ​​​​​​​

= 0.0643

(b)

(Sum of infinite Geometric series with first term as a and common ratio r = a/(1-r) )

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