Question

In a horse race with 5 horses, you make a bet by predicting the ranking of all 5 horses. Suppose you place your bet at random. What is the probability that you will rank the top 2 horses incorrectly in the incorrect order? Write your answer as an exact fraction which is reduced as much as possible.

Answer 1

Answer:

P( top two horses are predicted incorrectly in incorrect order)

= $\frac{1}{2}$

Step-by-step explanation:

In the horse race the outcome can be predicted in 5! = 120 ways.

Now suppose the top two horses were predicted incorrectly in incorrect order. Now, the  top horse can be predicted incorrectly in 4 ways.

Suppose the top horse was predicted to be in k-th position where k = 2, 3 ,4,5

so the second horse can be predicted to be in place from 1 to (k - 1)

So, the top two horses can be predicted  incorrectly in incorrect order

in $\sum_{k =2}^{5}(k - 1)$ = 10 ways  

and for each prediction of the two the remaining horses may be predicted in 3! = 6 ways.

Hence ,

P( top two horses are predicted incorrectly in incorrect order)

= $\frac{6 \times 10}{120}$

=$\frac{1}{2}$