Horse Racing odds question

[JellyFish]

Laura and Neil are racing horses. Laura's odds for winning are 2 to 1, the odds in favour of either Neil or Laura winning are 3 to 1. What are the smallest and the largest possible values of the probability that Neil wins the race?
Any help would be greatly appreciated. I'm really uncertain how to begin this problem.

 

[soroban]

Hello, JellyFish!

The problem is strange . . .

Laura and Neil are racing horses.
Laura's odds for winning are 2 to 1.
The odds in favour of either Neil or Laura winning are 3 to 1.
What are the smallest and the largest possible values .??
of the probability that Neil wins the race?

\(\displaystyle \text{We are given: }\;\begin{array}{ccc}P(L) &=& \frac{2}{3} \\ \\[-3mm] P(L \vee N) &=& \frac{3}{4}\end{array}\)


\(\displaystyle \text{I would assume that there is just }one\text{ winner . . . There are no ties.}\)

. . \(\displaystyle \text{That is: }\(L \wedge N) \:=\:0\)


\(\displaystyle \text{Then we have: }\;\underbrace{P(L \vee N)}_{\frac{3}{4}} \;=\;\underbrace{P(L)}_{\frac{2}{3}} + P(N) - \underbrace{P(L \wedge N)}_{0}\)


\(\displaystyle \text{Therefore: }\;\frac{3}{4} \;=\;\frac{2}{3} + P(N) \quad\Rightarrow\quad P(N) \:=\:\frac{1}{12}\)

 

[JellyFish]

Thanks Soroban, I knew I wanted to use that equation to solve for P(N) but I guess I don't really understand the odds of horse racing to get 2/3 and 3/4, so I thank you very much.

 

[JellyFish]

I think that the way that Soroban answered the question is the smallest probabilit6y for Neil winning because the odds of 2 to 1 were used in favour of Laura.

If I use the odds of 1 to 2 against Laura then then P(L) = 1/3 and therefor P(N)= 5/12 which would be the largest probabilty of Neil winning. Do you think that works?