Playing a game at a large social event, Elaine pays $20 to claim three numbers from 1 through 20 without repetition. Near the end of the event, three different winning numbers will be drawn at random from 1 through 20. Elaine will receive $20 if she has exact one winning number, $50 if she has exact two winning numbers, $200 if she has three winning numbers, and nothing if there is no match.   (a) Find Elaine’s expected net gain and the standard deviation of her net gain.   (b) Assume that 300 people played this game at the event. Find the expected net gain of the event organizer from this game assuming there is $200 fixed cost.

To raise money for the local Veterans Affairs hospital, your friend organizes a fundraiser, inviting you to play a two-stage game where you pay $8 to play. The game works as follows: a fair 8-sided die is rolled, noting the number shown, and a spinner divided into 4 equal regions of different colors (blue, red, green, orange) is spun, noting the color. If the die shows 3 or the spinner shows orange, then you win $21. If the die shows an even number and the spinner does not show orange, then you win $9. Otherwise, you do not win anything. Let X be your net winnings. (a) Create a probability distribution for X. Enter the possible values of X in ascending order from left to right. All probabilities should be exact. X       P(X)         (b) Compute your expected net winnings for the game. Round your answer to the nearest cent. $    (c) Is this game fair? Yes No

At a county fair, you pay $2 to play the following game: You are given a fair coin, and two bags standin front of you, labelled Bag A and Bag B. You flip a coin: If your coin lands Heads, you pick a ballfrom Bag A. If your coin lands tails, you pick a ball from Bag B. Each ball has a dollar value written onit, and you win the number of dollars shown on the ball. • Bag A contains four (4) balls in total: Three (3) balls show $1, and one (1) ball shows $5.• Bag B contains three (3) balls in total: Two (2) balls show $2, and one (1) ball shows $3.You WIN money if you leave with more money than you started with. You LOSE money if you leavewith less money than you started with. You BREAK EVEN if you leave with the same money that youstarted with.Let H denote the event that you flip heads, T denote the event that you flip tails, W the event that youwin, and L the event that you lose.(i) You pay to play exactly one game. What is the probability that you WIN given thatyou flipped a Heads?(ii)…

A lawyer, considering taking a case figures that there is a two and five chances of winning the case in the court and earning his firm $400,000, a two and five chances of losing and costing his firm $250,000, and a one and five chances of settling out of court and bringing in $200,000. Should the lawyer take the case based on his expected payoff? explain using mathematical argument.