One die and 1, 2 or 3 coins

Probability Level 3

A fair die is tossed.
If the die lands on 1, 2 or 3, then 3 fair coins are tossed.
If the die lands on 4 or 5, then 2 fair coins are tossed.
If the die lands on 6, then 1 fair coin is tossed.

Given that the resulting coin tosses produced no tails, what is the probability that the die landed on 6?

The probability can be expressed as a b \dfrac ab for coprime positive integers a a and b b , find the value of a + b a+b .

Image Credit: Flickr nickphotos .


The answer is 15.

Paul Fournier by Paul Fournier , 70, Canada

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A = probability the die is '6' AND no tails
A = (1/6)(1/2) = 1/12

B = probability of no tails

Case 1: die lands on 1,2, or 3 and no tails: (3/6)(1/2)^3
Case 2: die lands on 4 or 5 and no tails: (2/6)(1/2)^2
Case 1: die lands on 6 and no tails: (1/6)(1/2)

B = (3/6)(1/2)^3 + (2/6)(1/2)^2 + (1/6)(1/2) = 11/48


A/B = (1/12) / (11/48) = 4/11

4 + 11 = 15

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