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The Monty Hall Problem explained

The Monty Hall Problem explained

(2:48) Probability puzzle that still stumps some mathematicians. Although it may seem counterintuitive that switching should make any difference, if you are faced with the same scenario, you should choose switch. No guarantee you'll win but you'll double your chances.

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Guest: Guest God (1834 days ago)
What happens if I start with two doors and pick one then add an open/empty door and make it 3 doors to choose from where I know the added door is empty? This is the reverse of the same scenario and your chance remain 50/50.
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What happens if I start with two doors and pick one then add an open/empty door and make it 3 doors to choose from where I know the added door is empty? This is the reverse of the same scenario and your chance remain 50/50.
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Guest: Goatherd (1834 days ago)
I want a goat. What should I do?
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I want a goat. What should I do?
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cengland0 cengland0 (1835 days ago)
Besides, past history does not indicate the probability of future events. For example, if I flipped a coin and it came up heads 10 times in a row, what are the chances that I flip it again and it will become heads. You might think it's a small chance but it's still 50/50. So whatever happened before one of the doors were opened, you still have two doors to pick from and one contains a car. That's a 50/50 chance.
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Besides, past history does not indicate the probability of future events. For example, if I flipped a coin and it came up heads 10 times in a row, what are the chances that I flip it again and it will become heads. You might think it's a small chance but it's still 50/50. So whatever happened before one of the doors were opened, you still have two doors to pick from and one contains a car. That's a 50/50 chance.
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TheBob TheBob (1834 days ago)
Who was it said that if you flip a coin and get heads 10 times in a row, it's likely to be heads on the 11th? This is because there's probably something wrong with the coin, or you wouldn't get 10 in a row. I know this is nothing to do with goats
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Who was it said that if you flip a coin and get heads 10 times in a row, it's likely to be heads on the 11th? This is because there's probably something wrong with the coin, or you wouldn't get 10 in a row. I know this is nothing to do with goats
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cengland0 cengland0 (1834 days ago)
First, I didn't say that the 11th toss would be more likely to be heads. I stated the chance is still 50/50. This is assuming the coin is weighted exactly the same for heads and tails and there are no defects in the coin and the coin toss is considered random and not rigged where an expert can select the number of flips for each toss -- controlling the outcome. If you flip a coin a trillion times, there are potentially several sequences where there are 10 heads in a row. Besides, the coin flipping analogy was to illustrate a sequence of past random events and not to illustrate the real randomness of a coin toss. I suppose if I knew a better random generator that was perfect, I would have used that in my discussion instead but I do not know of any.
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First, I didn't say that the 11th toss would be more likely to be heads. I stated the chance is still 50/50. This is assuming the coin is weighted exactly the same for heads and tails and there are no defects in the coin and the coin toss is considered random and not rigged where an expert can select the number of flips for each toss -- controlling the outcome. If you flip a coin a trillion times, there are potentially several sequences where there are 10 heads in a row. Besides, the coin flipping analogy was to illustrate a sequence of past random events and not to illustrate the real randomness of a coin toss. I suppose if I knew a better random generator that was perfect, I would have used that in my discussion instead but I do not know of any.
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TheBob TheBob (1834 days ago)
That's exactly the point. IF (Capital "I", capital "F") it comes up heads 10/10 times it's more likely to be a "fixed" coin (or a dodgy tosser) than a random occurrence of 1/1024. Therefore toss #11 is unlikely to be 50/50. I'm not arguing with you, I was just wondering if anyone knew who came up with this idea.
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That's exactly the point. IF (Capital "I", capital "F") it comes up heads 10/10 times it's more likely to be a "fixed" coin (or a dodgy tosser) than a random occurrence of 1/1024. Therefore toss #11 is unlikely to be 50/50. I'm not arguing with you, I was just wondering if anyone knew who came up with this idea.
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cengland0 cengland0 (1834 days ago)
My statistics teacher used that as an example when I went to college. That class was taken in 1985, I believe. So it must have been an idea "tossed" around (pun intended) prior to 1985.
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My statistics teacher used that as an example when I went to college. That class was taken in 1985, I believe. So it must have been an idea "tossed" around (pun intended) prior to 1985.
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Guest: Guest101 (1834 days ago)
Nope. The doors do not give a 50/50 chance.
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Nope. The doors do not give a 50/50 chance.
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Guest: Mr Hall (1834 days ago)
cengland - Past history is indeed no predictor of future events - but only if they are independent events. In the MH problem they are not independent.
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cengland - Past history is indeed no predictor of future events - but only if they are independent events. In the MH problem they are not independent.
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Guest: Kamikaze (1834 days ago)
The whole point of the analysis is thatg in thins case, past history DOES matter, since the objects behind rhe doors don't move. If the doors and their contents were randomized after the one door was eliminated, there would indeed be a 50% chance of guessing right. However, since the doors are not reset, revealing an incorrect door does not change the fact that your first guess was probably wrong.
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The whole point of the analysis is thatg in thins case, past history DOES matter, since the objects behind rhe doors don't move. If the doors and their contents were randomized after the one door was eliminated, there would indeed be a 50% chance of guessing right. However, since the doors are not reset, revealing an incorrect door does not change the fact that your first guess was probably wrong.
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Guest: Surrey_Tidal_Wave (1835 days ago)
It is flat out wrong and shows how 'science' get things wrong it is common sense that no difference is made when you pick door. Duh.
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It is flat out wrong and shows how 'science' get things wrong it is common sense that no difference is made when you pick door. Duh.
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Guest: guestme (1833 days ago)
Troll
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Troll
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Guest: Guest101 (1834 days ago)
Troll.
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Troll.
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danskal danskal (1829 days ago)
Latest comment: Behind one of the above comments is a troll. Would you pick this one, or would you prefer to switch? Yeah, me too.
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Latest comment: Behind one of the above comments is a troll. Would you pick this one, or would you prefer to switch? Yeah, me too.
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cengland0 cengland0 (1835 days ago)
Wait a second, this may not be the way it works. If you picked door #1 and it contained the goat, they might just say you lost and not give you a chance to switch. The fact that you did pick the door the car was behind makes them want you to switch so they will do whatever they can to make you switch including showing you one of the doors with a goat. If they always gave you the option to switch regardless if your initial pick had the car or a goat, then the math would always be a 50/50 chance. That's because regardless of what door you pick in the beginning, there will always be one with a goat that you did not pick. They will then reveal which one that is and then allow you to pick again with the remaining two doors. That gives you a 50/50 chance but this video is trying to confuse people about the odds.
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Wait a second, this may not be the way it works. If you picked door #1 and it contained the goat, they might just say you lost and not give you a chance to switch. The fact that you did pick the door the car was behind makes them want you to switch so they will do whatever they can to make you switch including showing you one of the doors with a goat. If they always gave you the option to switch regardless if your initial pick had the car or a goat, then the math would always be a 50/50 chance. That's because regardless of what door you pick in the beginning, there will always be one with a goat that you did not pick. They will then reveal which one that is and then allow you to pick again with the remaining two doors. That gives you a 50/50 chance but this video is trying to confuse people about the odds.
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Guest: PeterLondon (1834 days ago)
cengland0 - on other posts here you boast about how many years you have been in the banking industry and you know all about how the economy and finance works!! HA HA. You can't even understand a simple game show on TV. No wonder the USA is going down the toilet.
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cengland0 - on other posts here you boast about how many years you have been in the banking industry and you know all about how the economy and finance works!! HA HA. You can't even understand a simple game show on TV. No wonder the USA is going down the toilet.
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cengland0 cengland0 (1834 days ago)
Did you hear him say that this scenario has perplexed scientists and mathematicians to this day. So if that's true and I'm wrong about this, then I feel good about it because scientists and mathematicians are perplexed about this. According to the video, computer simulations (and I wrote one to verify and will share it if anyone is interested) confirm the chances are greater if one door is ALWAYS revealed and you switch. However, if you pick the wrong door and they immediately tell you that you lost, then what are your chances? So I'll humbly agree that it's not 50/50 if you can always switch. My computer simulation shows 312 Wins, 688 Losses (31% wins) when you do not switch for 1000 plays. If you do switch, I get 654 Wins, 346 Losses (65% wins). I tried this multiple times: 2nd attempt (673 Wins, 327 Losses), 3rd attempt (687 Wins, 313 Losses), 4th attempt (671 Wins, 329 Losses).
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Did you hear him say that this scenario has perplexed scientists and mathematicians to this day. So if that's true and I'm wrong about this, then I feel good about it because scientists and mathematicians are perplexed about this. According to the video, computer simulations (and I wrote one to verify and will share it if anyone is interested) confirm the chances are greater if one door is ALWAYS revealed and you switch. However, if you pick the wrong door and they immediately tell you that you lost, then what are your chances? So I'll humbly agree that it's not 50/50 if you can always switch. My computer simulation shows 312 Wins, 688 Losses (31% wins) when you do not switch for 1000 plays. If you do switch, I get 654 Wins, 346 Losses (65% wins). I tried this multiple times: 2nd attempt (673 Wins, 327 Losses), 3rd attempt (687 Wins, 313 Losses), 4th attempt (671 Wins, 329 Losses).
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cengland0 cengland0 (1834 days ago)
Yes I am a banker and you're also assuming the game show ALWAYS lets you switch.
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Yes I am a banker and you're also assuming the game show ALWAYS lets you switch.
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Guest: PeterLondon (1833 days ago)
I'm assuming you can watch a simple video and not infer events or choices which are clearly not present in the video. You are indeed right that this scenario has perplexed scientists and mathematicians to this day. But interestingly, if you give the problem to a statistician who specialises in probability theory, then they will tell you what the odds are straight-away. That's because they specialise in this specific area of mathematics. Just because someone is a "scientist" - possibly specialising in dinosaur poo - or a "mathematician" doesn't mean they will be any better in this specific area then any other reasonably well-educated person. But you have "humbly agreed" that the video is correct - so "respect" as the man says.
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I'm assuming you can watch a simple video and not infer events or choices which are clearly not present in the video. You are indeed right that this scenario has perplexed scientists and mathematicians to this day. But interestingly, if you give the problem to a statistician who specialises in probability theory, then they will tell you what the odds are straight-away. That's because they specialise in this specific area of mathematics. Just because someone is a "scientist" - possibly specialising in dinosaur poo - or a "mathematician" doesn't mean they will be any better in this specific area then any other reasonably well-educated person. But you have "humbly agreed" that the video is correct - so "respect" as the man says.
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glortman glortman (1833 days ago)
I admire cengland0's courage for being honest about his profession, especially knowing how people trash each other on sites like this. There are lots of roles in banking and only investment and lending require any knowledge of probability, which has its own math. The MHP is very complex: I have a friend who teaches a graduate level math course on it and other related problems. In his view, the issue is psychological: more about people's beliefs than it is about the numbers, since many of his students when presented with clear proof, will still reject the correct answer.
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I admire cengland0's courage for being honest about his profession, especially knowing how people trash each other on sites like this. There are lots of roles in banking and only investment and lending require any knowledge of probability, which has its own math. The MHP is very complex: I have a friend who teaches a graduate level math course on it and other related problems. In his view, the issue is psychological: more about people's beliefs than it is about the numbers, since many of his students when presented with clear proof, will still reject the correct answer.
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cengland0 cengland0 (1832 days ago)
Thank you for the kind words. Before I believe what people tell me, I have to verify it for myself. So I did write a computer simulation and it verifies what the video says. I'm still surprised but it is what it is.
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Thank you for the kind words. Before I believe what people tell me, I have to verify it for myself. So I did write a computer simulation and it verifies what the video says. I'm still surprised but it is what it is.
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Guest: PeterLondon (1833 days ago)
glortman - you are correct. The MHP is both very complex and simple at the same time (once you get it). Indeed the issue is psychological, but not so much because of people's beliefs - rather their simple lack of understanding of what is going on as the show host is not randomly selecting one door out of two to remove. I have found the easiest way to grasp the MHP is this: When the host selects a door to remove they are TELLING you something. 2 times out of 3 they are telling you that the door you did not select is the one with the car behind it. And 1 time out of 3 they are telling you it has a goat behind it. So you should always swap.
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glortman - you are correct. The MHP is both very complex and simple at the same time (once you get it). Indeed the issue is psychological, but not so much because of people's beliefs - rather their simple lack of understanding of what is going on as the show host is not randomly selecting one door out of two to remove. I have found the easiest way to grasp the MHP is this: When the host selects a door to remove they are TELLING you something. 2 times out of 3 they are telling you that the door you did not select is the one with the car behind it. And 1 time out of 3 they are telling you it has a goat behind it. So you should always swap.
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glortman glortman (1832 days ago)
I like your analysis, PL.
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I like your analysis, PL.
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Guest: Kamikaze (1834 days ago)
True, the game show could be pulling a fast one on you. However, the puzzle assumes the game show is honest, and that you are given the choice regardless of whether you initially chose right or wrong.
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True, the game show could be pulling a fast one on you. However, the puzzle assumes the game show is honest, and that you are given the choice regardless of whether you initially chose right or wrong.
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glortman glortman (1833 days ago)
Assuming a fair coin, EVERY toss has a 50% chance of being heads or tails, regardless of the prior tosses. Each individual probability is independent of all of the others: 1/2, and no number of tosses will influence any future tosses. Even if a fair coin was tossed 10 times heads, the next toss would still have a 50% chance of coming up tails. The chances of 10 successive heads tosses diminishes with each toss, and is 1/1024, as theBob points out. The events in this second instance are not independent. So they are two different probabilities. The MHP is an instance of the latter. The events are NOT independent. The game show hosts remove a choice they KNOW is not correct, and reveal that to the player: 2/3 chance they initially chose the wrong door. Imagine the player was just shown the removal of one wrong choice before he chose anything. In this instance, there is a 50% chance of getting the right one.
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Assuming a fair coin, EVERY toss has a 50% chance of being heads or tails, regardless of the prior tosses. Each individual probability is independent of all of the others: 1/2, and no number of tosses will influence any future tosses. Even if a fair coin was tossed 10 times heads, the next toss would still have a 50% chance of coming up tails. The chances of 10 successive heads tosses diminishes with each toss, and is 1/1024, as theBob points out. The events in this second instance are not independent. So they are two different probabilities. The MHP is an instance of the latter. The events are NOT independent. The game show hosts remove a choice they KNOW is not correct, and reveal that to the player: 2/3 chance they initially chose the wrong door. Imagine the player was just shown the removal of one wrong choice before he chose anything. In this instance, there is a 50% chance of getting the right one.
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Guest: Student (1833 days ago)
given a coin that has just come up heads 10 times in a row, what is the probability that the coin is fair? Under the null hypothesis (ie that the coin is fair) the probability of 10 heads is 1 in 1024. That's low enough to reject the null hypothesis in an experimental scenario. But one experiment is seldom convincing. I'd want to see some more tosses before I bet the farm on another head.
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given a coin that has just come up heads 10 times in a row, what is the probability that the coin is fair? Under the null hypothesis (ie that the coin is fair) the probability of 10 heads is 1 in 1024. That's low enough to reject the null hypothesis in an experimental scenario. But one experiment is seldom convincing. I'd want to see some more tosses before I bet the farm on another head.
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glortman glortman (1832 days ago)
You have changed the question slightly, and it has important consequences. The two questions so far: What is the probability of heads on any flip? and What is the probability of 10 flips in a row? are subject to simple probability rules related to the degrees of freedom in any one flip, or 10 successive flips. Your new question: What is the probability the coin is fair? is of a different nature, and you need to use a Chi-Square to determine a frequency distribution of some large number of coin flips, not simply examining ten heads flips in a row. The Chi-Square considers the frequency of observed over expected results, but even then does not consider the number of times in a row the observed happens, since that is subject to chance.
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You have changed the question slightly, and it has important consequences. The two questions so far: What is the probability of heads on any flip? and What is the probability of 10 flips in a row? are subject to simple probability rules related to the degrees of freedom in any one flip, or 10 successive flips. Your new question: What is the probability the coin is fair? is of a different nature, and you need to use a Chi-Square to determine a frequency distribution of some large number of coin flips, not simply examining ten heads flips in a row. The Chi-Square considers the frequency of observed over expected results, but even then does not consider the number of times in a row the observed happens, since that is subject to chance.
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Guest: (1835 days ago)
cenglandO you are a fool. how you can question this amazes me. your first choice is 1 in 3. those odds do not change when you remove incorrect choices. you still made your choice when there were three options present. only when you switch do you actually make a choice of odds 1 in 2. staying with the first choice you stay with the odds of 1 in 3. he writes out the math in the video. Just because you cant do the math doesn't mean he is trying to confuse people.
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cenglandO you are a fool. how you can question this amazes me. your first choice is 1 in 3. those odds do not change when you remove incorrect choices. you still made your choice when there were three options present. only when you switch do you actually make a choice of odds 1 in 2. staying with the first choice you stay with the odds of 1 in 3. he writes out the math in the video. Just because you cant do the math doesn't mean he is trying to confuse people.
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Guest: Kamikaze (1834 days ago)
Ok, how about an even simpler breakdown? 1) Pick a door. Your chances of being correct are 1/3. 2) I inform you that at least 1 of the doors you did not choose was incorrect. Did I really tell you anything? With three doors, there will ALWAYS be one incorrect door that you did not pick, so this information changes nothing. Opening a door to prove it changes nothing. This does not change the fact than any time your first choice was INCORRECT, you win by switching doors. PERIOD. 3) If you dispute the resuts, get a friend and TRY IT!!!!!! You'll find that switching doors wins more often than not switching.
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Ok, how about an even simpler breakdown? 1) Pick a door. Your chances of being correct are 1/3. 2) I inform you that at least 1 of the doors you did not choose was incorrect. Did I really tell you anything? With three doors, there will ALWAYS be one incorrect door that you did not pick, so this information changes nothing. Opening a door to prove it changes nothing. This does not change the fact than any time your first choice was INCORRECT, you win by switching doors. PERIOD. 3) If you dispute the resuts, get a friend and TRY IT!!!!!! You'll find that switching doors wins more often than not switching.
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