In one version of the game Keno the house has a pot containing 80 balls numbered 1 through 80,.the house then selects 20 of the 80 numbers at random .

The house randomly selects 20 numbers from the counting numbers 1-80 .In the variation called 6-spot keno, the player pays 60 cents for his ticket and marks 6 numbers of his choice .If the 20 numbers selected by the nouse countain at least 3 of those chosen by the player, he gets a payoff according to this scheme.
3 of the players "s numbers among the 20 ----- $0.35
4 of the player's numbers among the 20 ------ $2.00
5 of the players numbers among the 20 ------ $60.00
6 of the player"s numbers among the 20 ------ 1250.00
Find the players expected net winnings in this game
Pleaeease show me the work

To find the player's expected net winnings in this game, we need to calculate the probability of each possible outcome and then multiply it by the corresponding payoff, and finally, subtract the cost of the ticket.

1. Calculating the probability:
- There are a total of 80 numbers, and the player chooses 6 of them. Therefore, the number of possible combinations of the player's numbers is given by the binomial coefficient C(80, 6). This can be calculated using the formula:
C(n, r) = n! / (r! * (n - r)!)
where n is the total number of items and r is the number of items chosen.

- The house selects 20 numbers from the 80, and we need to find the probability of getting exactly 3, 4, 5, or 6 matches with the player's numbers. This can be calculated using the probability mass function for hypergeometric distribution, which is given by the formula:
P(X = k) = (C(m, k) * C(N-m, n-k)) / C(N, n)
where N is the total number of items, n is the number of items chosen, m is the number of "good" items, and k is the number of successful outcomes.

2. Calculating the expected net winnings:
- We multiply the probability of each outcome by its corresponding payoff to find the expected value. Given the payoffs mentioned in the question, we get:
E(net winnings) = (P(X = 3) * $0.35) + (P(X = 4) * $2.00) + (P(X = 5) * $60.00) + (P(X = 6) * $1250.00)

3. Subtracting the cost of the ticket:
- The player pays 60 cents for the ticket, so we subtract this cost from the expected net winnings.

By following these steps, we can determine the player's expected net winnings in this game.