I 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) can the math be shown on how to do the combinations and probability?

Solution

A) IN A DECK THERE ARE 52 CARDS

WE HAVE TO GET THE PROBABILITY OF WINNING $X - NOW

IT DEPENDS FOR THE NUMBER OF CARDS ER ARE CHOOSING

FOR N = 1 THE PROBABILITY OD WINNING WILL BE 1/52

FOR N=2 THE PROBABILITY OF WINNING WILL BE 1/52*1/51

FOR N=3 THE PROBABILITY OF WINNING WILL BE 1/52*1/51*1/50

SAME FOLLOWS FOR OTHER VALUE OF N

B)IF N = 4 THEN X = 270725

EXPECTED VALUE IS X1*P(X1)+X2*P(X2)....

WE HAVE TO EQUAL THE EXPECTED VALUE TO $1

NOW WE CAN HAVE X = 132,600

FOR N =3

P(3)= 1/52*1/51*1/50=1/132600

THEREFORE 270725/132600 = 2.04