Q) An unbiased die is rolled until a number greater than 4 appears on the die. Find the probability that an even number of tosses are needed.

A)
prob. that a number greater than 4 appears = 2/6 = 1/3
prob. that a number less than or equal to 4 appears = 2/3

prob. that an even no. of tosses are needed =
(2/3)*1/3 + (2/3)^3*1/3 + (2/3)^5 * 1/3 ......
= 2/9 (1 + (2/3)^2 + (2/3)^4 + (2/3)^6 + ....)
= 2/9 [ 1 / ( 5/9) ]
= (2/9)*(9/5)
= 2/5

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