Monty Hall Problems

On the classic television show Let's Make a Deal, a contestant is asked to choose one of three doors-with a grand prize behind one door and goats behind the other two doors. After the contestant picks a door, the host, Monty Hall, does what he always does by showing a goat behind a door that was not chosen and asking if the contestant wants to switch doors.

What do you advise?

What is wrong with the answer to this Car Talk Puzzler?

Monty Hall. Let's Make a Deal. Doors 1, 2, and 3... You select doors without knowing what is behind them. One of the doors has a good prize behind it, the others have crummy prizes. Monty says, "You picked door 1, but you don't know what's behind it. I've shown you what's behind door 2, and it is a bunch of stale Eskimo pies... Would you like to switch to door 3?" What do you do, and why? > Answer: The answer is, you should switch. Monty Hall is always going to show you a crummy door. Right? So you have a 50/50 chance of getting a good door, if you switch. So statistically, if you do this enough times, by switching doors, you up the chance of getting a good outcome.

There are four doors: one door with a $\$ 10,000$ prize hidden behind it and three doors with bags of goldfish. You choose Door 3. No matter which door you choose, the host will then show you a bag of goldfish behind a door you did not choose (say, Door 1) and ask if you wish to switch your choice to one of the two remaining doors. What is the probability of winning the $\$ 10,000$ prize if you do switch?

There are three boxes: a box containing two gold coins, a box containing two silver coins, and a box containing one gold coin and one silver coin. You choose a box randomly and choose a coin randomly from this box. If it is a gold coin, what is the probability that it came from the box with two gold coins?

A box has three drawers. One drawer contains two gold coins; another drawer contains two lead coins; and the third drawer contains one gold coin and one lead coin. You pick a drawer at random and a coin from that drawer at random. It is a gold coin, which is set aside and you do not get to keep. Now, your options are to either (a) take the other coin in the drawer that you chose; or (b) randomly pick a coin from one of the two drawers you did not choose. If you choose Option (a), what is the probability that you will get a gold coin?

Five coins are placed in a leather bag. One coin is heads on both sides; one coin is tails on both sides, and three coins are heads on one side and tails on the other side. A coin is drawn randomly from the bag and, without anyone looking at it, is flipped in the air. It lands with a heads side up. What is the probability that the other side of the coin is tails?

Three-card Monte: three cards are placed in a bag. One card is red on both sides, one is green on both sides, and one is red on one side and green on the other side. You draw a randomly selected card out of the bag and see that it is green on the front side. The operator of the game says, "We know that this is not the double-red card. We also know that it could be either green or red on the other side. I will bet $\$ 5$ against your $\$ 4$ that it is, in fact, green on the other side." Can the operator profit from such bets without cheating?

Two cards are placed in a leather bag. One card is red on both sides; the other card is red on one side and blue on the other side. If you pick a card randomly and look at a side randomly and see that it is red, what is the probability that the other side of the card is also red?

A wooden box has four cards inside. One card is red on both sides; one card is blue on both sides; and two cards are red on one side and green on the other side. If you pick a card randomly and look at a side randomly and see that it is red, what is the probability that the other side is green?

Three prisoners, A, B, and C, are in separate cells and sentenced to death. The governor has randomly selected one of them to be pardoned. The warden knows which one, but is not allowed to tell. Prisoner A begs the warden to let him know the identity of one of the others who will be executed. "If B is to be pardoned, give me C 's name. If C is to be pardoned, give me B's name. And if I'm to be pardoned, flip a coin to decide whether to name $B$ or $C$." The warden tells $A$ that $B$ is to be executed. Prisoner A is pleased because he believes that his probability of surviving has gone up from $1 / 3$ to $1 / 2$, as it is now between him and C. Prisoner A tells $C$ the news, who is also pleased because he reasons that A still has a $1 / 3$ chance of being pardoned, so his chance has gone up to $2 / 3$. What is the correct answer?

Mr. Smith is the father of two. We meet him walking along the street with a young girl who he proudly introduces as his daughter. What is the probability that Mr. Smith's other child is also a girl? (Assume that boys and girls are equally likely and independent.) Don't just guess. Prove your answer.

Answer this letter to newspaper columnist Marilyn vos Savant, who is listed in the Guinness Book of World Records Hall of Fame for "Highest IQ":

Three of us couples are going to Lava Hot Springs next weekend. We're staying two nights, and we've rented two studios because each holds a maximum of only four people. One couple will get their own studio on Friday, a different couple on Saturday, and one couple will be out of luck. We'll draw straws to see which are the two lucky couples.

I told my wife we should just draw once, and the loser would be the couple out of luck both nights. I figure we'll have a two-out-of-three (66 $2 / 3 \%$) chance of winning one of the nights to ourselves. But she contends that we should draw straws twice-first on Friday and then, for the remaining two couples only, on Saturday-reasoning that a one-in-three (33 1/3) chance for Friday and a one-in-two (50) chance for Saturday will give us better odds....

Which way should we go?

Answer this letter to Marilyn Vos Savant: "If I flip a pair of coins until at least one of them lands heads, what are the chances that the other coin also has landed heads?"