A state lottery requires you to choose 4 cards from an ordinary deck: I heart, 1 club, 1 diamond, and 1 spade in that order from the 13 cards in each suit. If all four choices are selected by the lottery, yo win $5,000. It costs $1 to play. Find the expected payback.
To find the expected payback, we need to calculate the probability of winning and losing.
Step 1: Calculate the probability of winning:
To win, we need to choose the correct 4 cards in the specified order. Since there are 13 cards in each suit, the probability of choosing the correct card in the first try is 1/13. For the second card, the probability is 1/12, since there are now only 12 cards left, and so on. Therefore, the probability of winning is (1/13) x (1/12) x (1/11) x (1/10) = 1/28680.
Step 2: Calculate the probability of losing:
Since there are a total of 52 cards in a deck and we need to choose 4 cards, the probability of losing is equal to the complement of winning, which is 1 - 1/28680 = 28679/28680.
Step 3: Calculate the expected payback:
The expected payback is calculated by multiplying the probability of winning by the amount won, and then subtracting the probability of losing multiplied by the cost to play. In this case, the amount won is $5000, and the cost to play is $1.
Expected payback = (1/28680) x $5000 - (28679/28680) x $1
Now we can calculate the expected payback:
Expected payback = ($5000/28680) - ($28679/28680)
Expected payback ≈ $0.1745
Therefore, the expected payback is approximately $0.1745.