Maths Word Problem - Statistics and Expected Value

Question

Paul selects two cards, at random and without

replacement, from a standard pack of cards.

What is the expected number of hearts that he

gets?

Answer 1

First find probability distribution for the number of hearts

0 hearts probability (39/52) x (38/51) [ there are 39 non-hearts ]

1 heart (13/52) x (39/51) x 2

2 hearts (13/52) x (12/51)

To find expected number, multiply these probabilities by the number of hearts and add

so

0 x (39/52) x (38/51) + 1 x 2 x (13/52) x (39/51) + 2 x (13/52) x (12/51) = (1x2x13x39 + 2 x 13 x 12)/ (52x51)

= 1326 / 2652 = 0.5

What this means in reality is that if you do this process N times where N is large (>1000) you would expect to get about 0.5 N hearts in total. (law of large numbers)

So if you did it 10000 times, you would get about 0.5 x 10000 = 5000 hearts in total