Racing odds
In a race the odds are: 2 to 1 against the rhinoceros winning and 3 to 2 against the hippopotamus winning. What are the odds against the elephant winning if the race is fair?
Age
11 to 14
Challenge level
Problem
Image
An elephant, a hippopotamus and a rhinoceros are entered for a
race. The odds for the rhinoceros and the hippopotamus are:
2 to 1 against the rhinoceros winning and
3 to 2 against the hippopotamus winning.
What are the odds against the elephant winning if the race?
Editor's note: The hint explains 2 to 1 against in fractional form
Getting Started
2 to 1 against the rhinoceros winning means the probability of it
winning is 1/3 and 2/3 of it losing (2+1=3).
Student Solutions
This is a nice solution from Andrei of School 205 Bucharest. Thank you Andrei.
Looking at the first race.
If the odds are 2 to 1 against the rhinoceros, then the probability that the rhinoceros will win is 1/(1 + 2) = 1/3. For the hippopotamus, the probability that he will win is 2/(2 + 3) = 2/5.
The total probability (that there is a winner) is 1. So, the probability that the elephant will win is:
1 -1/3 -2/5 = 4/15
So, the odds for him are 11 to 4 against the elephant.