Homework SourseI have a standard deck of playing cards with 52 cards Suppos
I have a standard deck of playing cards with 52 cards. Suppose that I let you guess n cards, and if those are the first n cards that I pick from the shuffled deck, then I pay you $X. We will assume the the order doesn’t matter, for example if n = 2 and you guess Jack of Diamonds and Ace of Spades, then as long as I draw those two cards, it doesn’t matter what order they show up. To play this game, you have to pay me $1. For each value of n = 1, 2, 3, 4, 5, 53, 51, 52, answer the following questions, (a) What is the probability of winning $X? (b) How high would X have to be to make your expected value of the bet equal to the cost of participating ($1)? (Hint: If n = 4, then X = $270, 725)
Solution
the following table shows the the probability of winning $X and the amount high that would X have to be to make expected value of the bet equal to the cost of participating $1
| n | combinations | Probability | X- Amount high to make expected value of the bet equal to the cost of participating $1 |
| 1 | 52 | 0.019230769 | $ 52 |
| 2 | 1326 | 0.000754148 | $ 1,326 |
| 3 | 22100 | 4.52489E-05 | $ 22,100 |
| 4 | 270725 | 3.69379E-06 | $ 270,725 |
| 53 | 0 | 0 | $ - |
| 51 | 52 | 0.019230769 | $ 52 |
| 52 | 1 | 1 | $ 1 |
