Suppose there are two full bowls of cookies. Bowl #1 has 10 chocolate chip and 20 oatmeal, while bowl #2 has 20 chocolate chip cookies and 10 oatmeal cookies. Our friend Bob picks a bowl at random, and then picks a cookie at random. We may assume there is no reason to believe Bob treats one bowl differently from another, likewise for the cookies. The cookie turns out to be an oatmeal cookie. How probable is it that Bob picked it out of Bowl #2?

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X 4. A bag of fast-food sandwiches contains two hamburgers, four cheeseburgers, and three chicken burgers. If Wyatt ordered a cheeseburger and a chicken burger, what is the probability that a) the first sandwich selected will be one of his choices? b) neither of the first two sandwiches will be among Wyatt's choices?

Suppose you have three bowls, X, Y , and Z with some number of red and blue balls. Bowl X has 2 red and 2 blue balls, bowl Y has 1 red and 3 blue balls, and bowl Z has 3 red and 1 blue ball. A bowl is chosen at random and a ball is drawn from it. What is the probability that the ball was drawn from bowl X given the ball was blue?

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Suppose you roll 2 dice. What is the pobability that the sum of the 2 dice is exactly 4, given that the 1st dice is a 1?

One package contains 5 blue, 4 red and 3 green marbles. We pay $ 14 to draw two marbles at random and without replacement. If we get two blue marbles a prize of $ 20 is won, with two red marbles a prize of $ 30 is won, with two green marbles a prize of $ 30 is won, with two green marbles a prize of $ 50 is won and with any other result they lose the $ 14 they paid to play. Find the expected value of the win in this game. Write your answer in dollars and cents without the $ sign Example: -5.06

250 adults are practicing every day before taking part in a high-stakes children's game. The game requires them carefully to remove a shape- either a star or an umbrella-from a sugar wafer. 30% of those who choose stars on a given day will switch to umbrellas the next day, and 20% of those who choose umbrellas on a given day will switch to stars the next day. (a) If 150 contestants practice with stars today, how many will practice with umbrellas tomorrow? (b) If the actual game is held a long time from now, what proportion of contestants do we expect will be practicing with stars each day close to then?

Bowl A contains 2 red chips; bowl B contains two white chips; and bowl C contains 1 red chip and 1 white chip. A bowl is selected at random, and one chip is taken at random from that bowl.

A three-finger Morra game is a game in which two players simultaneously show one, two, or three fingers at each round. The outcome depends on a predetermined set of rules. Here is an interesting example: If the numbers of fingers shown by A and B differ by 1, then A loses one point. If they differ by more than 1, the round is a draw. If they show the same number of fingers, A wins an amount equal to the sum of the fingers shown.   Determine the optimal strategy for each player. (Enter your probabilities as fractions.) Player A should show one finger with probability _____ , two fingers with probability _____  , and three fingers with probability ______ . Player B should show one finger with probability _______  , two fingers with probability ________ , and three fingers with probability ________ .   Find the expected value of the game. The expected outcome is the player A will win __________ points per round, on average.