1. What was the probability that the third card they chose did in fact make three of a kind?3c6 = 20. Two combinations are winners. Ergo, 1 in 10.
2. If they could have switched to one of the other three cards, would that probability change, and if so, to what?Well, 1 in 4, ostensibly. Four cards left, one is right, three are wrong, they get to pick which one they want. As you said, they are shown their guaranteed pair, so everything up to that point is chrome.
3. Before revealing the third card the couple chose, the producers revealed a pair from the three unchosen cards (again, this was guaranteed to exist). If the couple were allowed to switch at that point and take the one other unrevealed card, how (if at all) would that affect their probability of winning the car?This is exactly the Monty Hall problem, and as such I won't touch it with a ten-foot pole. Really, (2) was closer than I'm comfortable talking about.
I can't say if 1/4 is wrong, but here's my take on Question #2:Your analysis of #2 is correct. And #3 is an even more blatant Monty Hall problem... the couple is 9 times more likely to win if they switch!
There is a 9/10 chance the contestant chooses to switch correctly. (The initial p = 1/10 is correct, so q = 9/10.) However, when they switch, there is only 1/3 chance the contestant is correct. Thus, wouldn't the probability here be 9/10 x 1/3 = 3/10?
I would agree that the last question fits the classic Monty Hall problem we are all familiar with.
Because I'm an alternate solution kind of guy, you could also go (1)(2/5)(1/4) = 1/10. After picking the first card, you have a 2-in-5 chance of matching on the second and a 1-in-4 chance of matching the third.1. What was the probability that the third card they chose did in fact make three of a kind?3c6 = 20. Two combinations are winners. Ergo, 1 in 10.
This is the conclusion I always arrive at with the Monty Hall Problem. The problem is, most people approach the Monty Hall Problem as if it were a casino game where the player has an unlimited number of trials. If that were the case then it is 100% true that the player should switch every time, giving himself a 66.6% chance of winning. However, if it is a true Monty Hall Problem within the context of a TV game show, the player has exactly one trial and the odds are as you stated above. The number of trials is less than the number of possible outcomes.2. If they could have switched to one of the other three cards, would that probability change, and if so, to what?Well, 1 in 4, ostensibly. Four cards left, one is right, three are wrong, they get to pick which one they want. As you said, they are shown their guaranteed pair, so everything up to that point is chrome.
This is the conclusion I always arrive at with the Monty Hall Problem. The problem is, most people approach the Monty Hall Problem as if it were a casino game where the player has an unlimited number of trials. If that were the case then it is 100% true that the player should switch every time, giving himself a 66.6% chance of winning. However, if it is a true Monty Hall Problem within the context of a TV game show, the player has exactly one trial and the odds are as you stated above. The number of trials is less than the number of possible outcomes.No matter how many times you say that, it still doesn't make any sense. If I only have one opportunity to play a game of chance, I am going to make the decisions that give me the best odds of winning that one game. Changing always improves your odds.
This is the conclusion I always arrive at with the Monty Hall Problem. The problem is, most people approach the Monty Hall Problem as if it were a casino game where the player has an unlimited number of trials. If that were the case then it is 100% true that the player should switch every time, giving himself a 66.6% chance of winning. However, if it is a true Monty Hall Problem within the context of a TV game show, the player has exactly one trial and the odds are as you stated above. The number of trials is less than the number of possible outcomes.I'm not following what you're saying. Do you mean to say that the (theoretical) probability of an outcome in a single trial shouldn't affect decision-making?
If you have a game that has a one-in-ten chance of winning a car, why would you increase those odds to one in four and one in two?To clarify, the switching questions I posed do NOT reflect what happened on today's show; they're simply hypothetical questions I posed similar to those in the original Monty Hall Problem.
Some of my favorite games from the old days were the stank-assed-luck, like finding $7 out of four envelopes with $1 and $2 bills, or Beat the Dealer, or the Cash Register, and on and on. Not a one of them ever had Monty reveal some part of the game and then play it over, and I thought all the better for it.
Chris, I agree with you that there is a practical difference between playing the game once and playing it a large number of times. Indeed, it's not that different from DoND, where even though the bank offer may be significantly less than the average of what's left, this is the one time the contestant is playing, and it's real money, so that affects their decision. However, it does not mean that the probabilities are any different. There may not be any long-term considerations if the player plays only once, but the probabilities don't change because of that.This is the conclusion I always arrive at with the Monty Hall Problem. The problem is, most people approach the Monty Hall Problem as if it were a casino game where the player has an unlimited number of trials. If that were the case then it is 100% true that the player should switch every time, giving himself a 66.6% chance of winning. However, if it is a true Monty Hall Problem within the context of a TV game show, the player has exactly one trial and the odds are as you stated above. The number of trials is less than the number of possible outcomes.2. If they could have switched to one of the other three cards, would that probability change, and if so, to what?Well, 1 in 4, ostensibly. Four cards left, one is right, three are wrong, they get to pick which one they want. As you said, they are shown their guaranteed pair, so everything up to that point is chrome.
They're playing some really good games from the sample size I've seen; so I can't understand why they'd water it down that way. (I like Panic Button as much as I loathe the game where a player has to roll 21 in 5d6.)
Chris, I agree with you that there is a practical difference between playing the game once and playing it a large number of times. Indeed, it's not that different from DoND, where even though the bank offer may be significantly less than the average of what's left, this is the one time the contestant is playing, and it's real money, so that affects their decision.To be clear, I agree with this as well. I'm simply saying (as everyone else is) that it doesn't change the underlying odds affecting a single trial.
Slightly OT: Are there clips of "Panic Button" anywhere? I watch LMAD from time to time but I've never seen this game show up.Beats me, I don't really seek out clips or episodes.
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.http://en.wikipedia.org/wiki/Law_of_large_numbers
The LLN is important because it "guarantees" stable long-term results for random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins.
My conclusion makes all kinds of sense if you are familiar with the Law of Large Numbers ...
Quote2. If they could have switched to one of the other three cards, would that probability change, and if so, to what?Well, 1 in 4, ostensibly. Four cards left, one is right, three are wrong, they get to pick which one they want. As you said, they are shown their guaranteed pair, so everything up to that point is chrome.
Same here. Certainly things will average out over time, and will not (in fact, cannot) exactly mirror the theoretical probabilities for the first case or cases. However, that does not change what the probability is... it only changes how you choose to interpret and use it.My conclusion makes all kinds of sense if you are familiar with the Law of Large Numbers ...
I'm familiar with the law of large numbers. I'm confused as to your using the law of large numbers and a simulation of thousands of trials to defend your position of "it's a one time shot." Maybe I'm mistaken.
The little games for a car that offered little chance to win (guess the suggested retail price of a can of paint on the West Coast within 50 cents, for example) got old even on Monty's version.Yes, but in those situations, people could make a somewhat educated guess.