Shawn is playing a game with a set of cards. Half of them are blue and the other half are yellow. In the game, the player guesses the color of the top card, looks at the card, and returns the card to the deck. The player continues to do this, shuffling the deck after each guess. Shawn has guessed the first three attempts as "blue” and has been correct on each guess. He says he will guess "yellow” for the next card since a yellow card is due to happen. Is Shawn’s reasoning correct? Yes, Shawn has guessed well so far, so it is clear that he cannot miss. No, all of the cards have been blue so far, so the next one must also be blue. Yes, half of the cards are yellow, so the fourth card should be yellow to compensate for the first three cards being blue. No, the law of large numbers says that the proportion of yellow cards should approach the true probability after many trials.

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

Sonic’s Drive-In Burgers had a Classic Car Cruise one night. They played oldies music, the servers wore roller skates, and there were 152 classic cars on display. Josie, the restaurant owner, played a game with the car owners. First everyone was told to turn their headlights on. Then Josie counted every other car (the 2nd, 4th, 6th, and so on), and told them to turn their headlights off. After that Josie counted every 3rd car (3rd, 6th, 9th, and so on). Those car owners were told to turn their headlights on If they were off, and turn their headlights off if they were on. She continued in this way, counting by every number (by 4’s, then by 5’s, then by 6’s, etc.) until she had finished counting by 152’s (which didn’t take long, since that’s how many cars there were.) As she counted, every car she pointed to reversed their lights from off to on or vice versa. When she was done, which cars still had their lights on?

Zara and Sue play the following game. Each of them roll a fair six-sided die once. If Sue’s number is greater than or equal to Zara’s number, she wins the game. But if Sue rolled a number smaller than Zara’s number, then Zara rolls the die again. If Zara’s second roll gives a number that is less than or equal to Sue’s number, the game ends with a draw. If Zara’s second roll gives a number larger than Sue’s number, Zara wins the game. Find the probability that Zara wins the game and the probability that Sue wins the game. Note: Sue only rolls a die once. The second roll, if the game goes up to that point, is made only by Zara.