If 8 equally qualified horses are running at the track and i

If 8 equally qualified horses are running at the track and if you are placing a single $1 bet and are choosing among the ensuing three wagers, determine which one would yield the greatest expected return; note that the expected return = - (bet amount) (lose probability) + (win amount) (win probability) where 0 < probability < 1.

6.1 Naming in order the first three horses to cross the finish line and win $1007

6.2 Naming in any order the first three horses to cross the finish line and win $167

6.3 Naming in any order at least one of the first three horses to cross the finish line and win $137

Solution

In case 6.1: probability of win is 1/ (8P3) {read as 8 permutations 3} since the total combination of top 3 horses can be in 8P3 ways (all equally likely since horses are equally qualified) and only 1 of which can be correct. So probability of losing would be [1- 1/(8P3) ].
1/(8P3) = 1/336- win probability and 335/336 is lose probability.
Hence expected return is equal to:

-(1)(335/336) + 1007(1/336) = 672/336= 2$

In case 6.2: probabiity of success will be 1/8C3 since total possible sets of top 3 horses from 8 eqully qualified horses can be chosen in 8C3 ways and we can predict only one of them right. 8C3= 1/56
probability of winning =1/56 and probability of losing = 55/56

Hence expected return=
-1(55/56)+ 167*(1/56)= 112/56= 2$

In case 6.3 probability of failure will be 5C3/8C3 since number of possible combinations of top 3 finishes will be 8C3 and not get not even at least one of them correct in our prediction will mean that none out of the predicted 3 horses finished in top 3 which means out of the rest 5 horses 3 horses finished top 3.
Failure probability= 5C3/8C3= 10/56 and success probability = 46/56

Expected return=
-1(10/56) + 137 (46/56)= 6292/56= 112.35$

Hence the greatest expected return would come in case 6.3