State yes/no in each part, and explain in a sentence or two. (a) You have a drawer filled with 20 balls, which are colored black or white. You have 12 black and 8 white balls. You randomly select a ball, record the color, and put the ball back into the drawer. Then you go to randomly select another from the drawer. Is the probability that the second pick is white independent of what the first ball color was? (b) You have a box containing 52 cards, which are either red or black. You have 26 red cards and 26 black cards. You randomly select a card (don't put it back in the box). Then you go to randomly select another card. Is the probability that the second pick is diamond independent of what the first card was? (c) A basketball player is going to do a free throw 100 times at practice. They are interested in the number of goals they make each kick is either a goal or not). If each free throw attempt is independent with a 90% chance (0.90 probability) of being a goal, then is the total number of free throws that are goals in the 100 attempts following a binomial distribution? (d) A soccer ball player is going to do 50 free kicks at practice. They are interested in the number of successes they make. If each kick is independent, but the player begins to make more free kicks the more attempts they make the probability they succeed increases the more attempts they take, P is increasing), then is the total number of successes made in the 50 attempts following a binomial distribution? (e) Your girlfriend has a box filled with scraps of paper, which have the names of movies (including the one you don't like). Your girlfriend randomly selects a scrap of paper from the box, reads it out loud, and sets the paper aside. Your girlfriend then randomly selects another paper. Is the probability that the second pick is the one you don't like independent of what the first picked movie was? (f) Your wife has a box filled with scraps of paper, which have the names of restaurants including the first place you went out with her). Your wife randomly selects a scrap of paper from the box, reads it out loud, puts the paper back in the hat, and mixes up the scraps of paper. Your wife then randomly selects another paper. Is the probability that the second pick is the first place you went out with her independent of what the first name was?

(a) You have a drawer filled with 20 balls, which are colored black or white. You have 12 black and 8 white balls. You randomly select a ball, record the color, and put the ball back into the drawer. Then you go to randomly select another from the drawer. Is the probability that the second pick is white independent of what the first ball color was?

yes. This is because they are independent events. The fact that you returned the first ball back to the drawer makes the initial conditions the same so the probability of picking a white ball is the same as in the first attempt.

(b) You have a box containing 52 cards, which are either red or black. You have 26 red cards and 26 black cards. You randomly select a card (don't put it back in the box). Then you go to randomly select another card. Is the probability that the second pick is diamond independent of what the first card was?

No. This is because the total number of cards in the box has changed since you didn't put the first card back. For example, the probability of picking a diamond on the first try is: P=13/52=1/4 = 25%. But when you pick the first card and don't put it back in the box, then the probability changes. If the first card was a card other than a diamond card, then the probability will now be: P=13/51=25.5% the probability has increased. If the first card you picked was a diamond, then the probability would be still different: P=12/51=23.5% so the probability of picking a diamond on the second attempt has decreased.

(c) A basketball player is going to do a free throw 100 times at practice. They are interested in the number of goals they make each kick is either a goal or not). If each free throw attempt is independent with a 90% chance (0.90 probability) of being a goal, then is the total number of free throws that are goals in the 100 attempts following a binomial distribution?

Yes. For it to be a binomial distribution you must only have two independent outcomes. In this case, the outcomes are the shot being a goal or not. The probability for it being a goal is 90% and the probability for it not being a goal is 10%. So it is a binomial distribution.

(d) A soccer ball player is going to do 50 free kicks at practice. They are interested in the number of successes they make. If each kick is independent, but the player begins to make more free kicks the more attempts they make the probability they succeed increases the more attempts they take, P is increasing), then is the total number of successes made in the 50 attempts following a binomial distribution?

No. This is because the outcomes of a binomial distribution must remain constant independently on the number of attempts. Since the probability increases with the number of outcomes, then the outcomes will not be independent so it becomes a hypergeometric distribution.

(e) Your girlfriend has a box filled with scraps of paper, which have the names of movies (including the one you don't like). Your girlfriend randomly selects a scrap of paper from the box, reads it out loud, and sets the paper aside. Your girlfriend then randomly selects another paper. Is the probability that the second pick is the one you don't like independent of what the first picked movie was?

No. This is because the number of possible outcomes will decrease as she sets the paper aside. If the first movie she picked was a movie you like, then the probability that the second movie she picks is the one you don't like increases. If the first movie she picked was the movie you don't like, then the probability that on the second try she picked the movie you don't like decreases to zero since it has alreay been picked.

(f) Your wife has a box filled with scraps of paper, which have the names of restaurants including the first place you went out with her). Your wife randomly selects a scrap of paper from the box, reads it out loud, puts the paper back in the hat, and mixes up the scraps of paper. Your wife then randomly selects another paper. Is the probability that the second pick is the first place you went out with her independent of what the first name was?

yes. This is because when she puts the paper back in the lot, then the total number of papers in the box is the same as in the first try. In this case the number of possible outcomes and the number of desired outcomes are the same as in the first try, so the probability is the same no matter what the first pick was.