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  4. Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The probability that Mr. A selected the winning horse is

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Q. Five horses are in a race. Mr. A selects two of the horses at random and bets on them. The probability that Mr. A selected the winning horse is

Probability

Solution:

Let 5 horses are $H_1, H_2, H_3, H_4$ and $H_5$. Selected pair of horses will be one of the 10 pairs (i.e.; ${^5C_2} ): H_1 H_2, H_1 H_3, H_1 H_4, H_1 H_5, H_2 H_3, H_2 H_4, H_2 H_5, H_3 H_4, H_3 H_5 $ and $H_4 H_5$.
Any horse can win the race in 4 ways.
For example : Horses $H_2$ win the race in 4 ways $H_1 H_2, H_2 H_3, H_2 H_4$ and $H_2 H_5$.
Hence required probability = $\frac{4}{10} = \frac{2}{5}$