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Andrew, Steven, and Adam are competing in a race. The odds of them winning the race were 5 : 3 : 2 5:3:2 respectively. (No draws)

Unfortunately, during the race, Andrew meets with an accident, which reduces his chances of winning to 1 3 \frac {1}{3} . What are the corresponding chances of winning for Steve and Adam now?

If Steven's probability becomes a b \frac {a}{b} and Adam's probability becomes c d \frac {c}{d} (all fractions simplified to lowest form), find the value of a + b + c + d a+b+c+d .


The answer is 26.

Parth Sankhe by Parth Sankhe , 27, India

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1 solution

Henry U
Dec 19, 2018

Andrew's reduced chanche doesn't change anything about Steven's and Adam's relative odds of winning, so they are still 3 : 2 3 : 2 . Furthermore, since there are no draws, the probability of either Steven or Adam winning is 1 − 1 3 = 2 3 1 - \frac 13 = \frac 23 . This probability is distributed in the ratio 3 : 2 3 : 2 among Steven and Adam, so their chances of winning are 2 3 ⋅ 3 3 + 2 = 2 5 \frac 23 \cdot \frac 3{3+2} = \frac 25 and 2 3 ⋅ 2 3 + 2 = 4 15 \frac 23 \cdot \frac 2{3+2} = \frac 4{15} respectively. Therefore, the answer is 2 + 5 + 4 + 15 = 26 2+5+4+15 = \boxed{26} .

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