Ok so today we are going to play a simple game. The game goes like this:
1. You are presented three doors.
2. You pick a door.
3. Another door is opened and you are given the chance to keep the door you picked or switch and select a new door.
What will you do? Wait what does mathematics tell you to do?
Task one: Tell us what you would do up front. Would you keep your original door or switch and select another door? Tell us why you chose to do what you did. Make sure to support your answer with mathematics!
Task Two: Go to the page below and play the game. Make sure to record your outcomes in the table provided on your webpage (right click here and open the web page in a new window or tab)
Game
http://www.shodor.org/interactivate/activities/SimpleMontyHall/?version=1.6.0_11&browser=Mozilla&vendor=Sun_Microsystems_Inc.
Task Three: Go the the Explore the Let's Make a Deal post below and tell us if you were right.
Good Luck and Have Fun!
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52 comments:
JW
I’m predicting that when you choose a door, and it shows you the door that doesn’t have a prize, you should switch from the door you have right now, to the door that hasn’t been chosen yet. I am predicting that your probabilty is higher then.
SV
I predict that staying with the original door will be the most likely to get you the prize.
When all three doors are unopened, you have a 33.3% chance of getting the car. After you choose one door, you have a 50% chance of getting the car in one door, and a 66.6% chance of getting the car in the other door. After choosing the second door, if you have not yet gotten the car, you have 100% chance of getting the car.
Mathematics tell you to switch doors after you pick the first one because is you switch you have a higher probability of getting the prize.
My prediction was incorrect. Switching doors is the best policy.
P.O.W.-------C.H.
This problem really depends on what is behind each door. If two doors are already opened then you have a 33.3% chance of getting something better or worse behind that third door. Say behind the three doors there is a 100 dollar prize behind one door, a 50 dollar prize behind another and a 25 dollar prize behind another. If you pick the 50 dollar prize then opening the second door there is a 33.3% chance of getting the 25 dollar prize and a 33.3% chance of getting the 100 dollar prize.
J.K. 2/13/09
I would switch doors. At the beginning there were three doors. I had a 1/3 chance of getting the correct door. Once a door was opened, there were two doors left. If I kept the same door, that would mean I still had a one-in-three chance of getting the door right. If I switched doors, I would have a one-in-TWO chance of getting it right because there were two doors when I re-chose my door. I would have a 50% chance of getting the door right rather than a 33.33%. This is why I would switch doors.
JAL
When you first start out with this game you have a 33.33 chance of picking the right door with the prize, so I’d start out with picking any door then once you pick a door, another one opens so now you now have only 2 doors to pick from, the one you started picking or the one that you haven’t picked and you have a 66.66 chance of picking the right door. So I would change doors from the first one I picked!
JE
I picked a door with 33.3 chance of getting the correct door.
But since you opened one door, u gave me a total of 66.6 chances
of winning and then I would switch for the prize.
S.S.
I would change doors. At first I have a 33.333% chance of picking the correct door. Once I have another door opened I will have 50% chance to pick the correct door. Once a door is opened I will either know if that’s the right door, or if it is a door not to pick. Therefore I can get it right right then and there, or have a 50% chance of winning still.
SV
I predict that staying with the original door will be the most likely to get you the prize.
When all three doors are unopened, you have a 33.3% chance of getting the car. After you choose one door, you have a 50% chance of getting the car in one door, and a 66.6% chance of getting the car in the other door. After choosing the second door, if you have not yet gotten the car, you have 100% chance of getting the car.
SV
I predict that staying with the original door will be the most likely to get you the prize.
When all three doors are unopened, you have a 33.3% chance of getting the car. After you choose one door, you have a 50% chance of getting the car in one door, and a 66.6% chance of getting the car in the other door. After choosing the second door, if you have not yet gotten the car, you have 100% chance of getting the car.
I would keep the door I chose. I am not sure how to explain why in mathematical reasons. I think it good to keep your first choice because first instinct is usually better. There is an example in my life where I chose something then later, another option opened. I turned it down and kept my first choice. I love him very much to this day. I would keep my first choice because I think it would be better odds to keep the first choice. After one of the doors is opened, there is only a 50% chance of picking the door with the good prize behind it.
JB POW
What I would do is I would stay with the door I already chose because you probably have a 33 percent chance on every door, and most of the time people will choose a different door and lose the game. So you probably have a better chance with the door you already chose.
POW MP
I think that I would keep the door I picked. Every door has a 33 percent chance of being the right one, and changing doors makes your chances even lower.
BJ
I would keep the door I chose. I am not sure how to explain why in mathematical reasons. I think it good to keep your first choice because first instinct is usually better. There is an example in my life where I chose something then later, another option opened. I turned it down and kept my first choice. I love him very much to this day. I would keep my first choice because I think it would be better odds to keep the first choice. After one of the doors is opened, there is only a 50% chance of picking the door with the good prize behind it.
AM
Once the second door was opened, if it was wrong, then that would mean that the chance of my door being correct was no 50% rather than 33%.
So, I would choose another door, for the sake of experiment. My prediction is that by choosing another door, it has a higher chance of being correct because the door that was wrong was opened, so then only one door is right. But the fact that that door was not opened shows that it might not be wrong. I would choose another door.
Either way, you end up with a fifty-fifty chance, but I would choose a different door.
If you pick a door and it opens another door, then it will change on the probability of what you will choose. If you choose and pick the same door that you picked at first, then you have a 1/3 chance of getting it right like how you started. However, if you pick and choose a different door, then you now have a ½ chance of getting the right door.
B.B.
The three different color doors give you a probability of 33% of winning what ever is behind the door. If I were to choose a door I would stick with the same color every time. I would do that because if you won with one of the colors, to keeps the luck going pick the same or similar color door and eventually one of the prizes will behind that door. If your look isn’t working switch off colors, pick the same color twice in a row to give you a sense of where the prize maybe placed. Also a good way of picking a door is to go with your gut.
T.G.
My prediction is that if I click one door and the pig comes up in an open door then I have a 50/50 chance of getting the prize. So I would probably stay with my own door because with a 50/50 chance I would want to stay with my gut feeling.
R.H
If I was presented with 3 doors and I had to choose just one, I would choose a random one because I would have a 33% chance I would have the winning door. Then when they open one of the doors, depending on the door, I would pick another door because now I would have a 50% chance of having the winning door.
C.C.
I wouldn’t stay with my original door because I only have about a 33% chance of winning. I would switch doors because then I would have a 50% chance of winning the game/ car. So I would actually have two out of three chances to win which is actually a 60% chance to win the game.
POW M.R.
If I could pick a new door after being presented three doors after being shown one of the doors I didn’t pick I would.
The reason for this is because on your first pick you had a 33% chance of picking the prize.
After the other door is shown you now have a 66% chance if you switch, the reason for this is because on your first door pick it was a 33% chance, so if you switch you will get the prize unless the first door you picked was the correct door, and since there is only a 33% chance of that it would make the most sense to switch doors.
CR
During this game I would most likely take the middle door so then one of the side doors open. With the three doors you have a one third of a chance of winning. When you open the first door, knowing it probably won’t be the prize, you now have a fifty percent chance of getting the prize. When you click on the door you have another door open and with that then you can decide if you want to keep or change door. I would probably keep my door because that’s my first guess and I am confident in my first choose.
POW NH
Once two doors are open you have a better chance of getting the better prize, but if you stick with your original choice than you have a less chance of getting the good prize. So if you only have one door open than you have a 33% chance of something good. With 2 doors open I have a 66% chance of a good prize. I would go for the other door to have a better chance of a good prize.
M.G
I would choose a random door out of the three, and a door will open with a pig in it. There are two doors left and I would change the door I picked earlier. At the beginning you have a 33.33 percent chance of picking the right door. After a door with a pig opens you have a 66.66 percent chance of getting the grand prize.
POW EK
I would keep the door I already picked. Each door has a 33 percent chance of being the right one. Asking if you want to switch is to make people feel unsure and switch, most of the time losing. When staying you have a better chance of wining
SV
Mathematics tell you to switch doors after you pick the first one because is you switch you have a higher probability of getting the prize.
Games stayed: 20
Games won: 6
Percent chance: 3%
Games switched: 20
Games won: 13
Percent chance: 6%
The data shows obviously that if you switch doors, you have a higher probability of winning.
My prediction was incorrect. I said staying would be better, but switching gives you a higher chance of winning.
POW R.B.
When you first open the door, you are going to get a pig that pops up at random. Then when that has past, I believe that you have a 50% chance of getting the prize or the pig.
JAL
My prediction was right!
I said that I would change doors. I did 65 games for each. For games stayed I did 65 games and won 13 of them to get a 20% chance of winning. Then I did 65 games of staying and won 42 of the games to get a 64.62% chance of winning!!!!!!!
As I said before you start out with a 33.3% chance of winning then once you have a 33.3% change of the car being in your door but a 66.6% chance of it being in the right door if you switch from your door to the other one.
AD
I think that I will choose a new door. You originally have a 33.33% chance of opening the right door. Once you choose your original door you have 66.66% chance of getting the right door if you switch doors. If you stay with your original door you only have a 50% chance of getting the prize. If you play 30 games and switch ever time you will win about 20 games. If you play 30 games an don’t switch at all you will win about 10 games.
SV
Mathematics tell you to switch doors after you pick the first one because is you switch you have a higher probability of getting the prize.
Games stayed: 20
Games won: 6
Percent chance: 33%
Games switched: 20
Games won: 13
Percent chance: 66%
The data shows obviously that if you switch doors, you have a higher probability of winning.
My prediction was incorrect. I said staying would be better, but switching gives you a higher chance of winning.
JML
At first when you pick a door you will have a 33.3% chance of winning the prize! So at first I would just pick any of the doors. After you choose a door, a door will open and obviously it will not contain the prize! So now you have a 50% chance of getting the prize but, in one of the doors you have a 33.3% of getting the prize and a 66.6% chance of getting the prize in the other door. I think that I would switch because you have the higher probability of getting the prize
AH
Task 1: Well, I would choose one of the other doors because there is 33% chance that the open door is correct or 66% incorrect. If the open one is incorrect, then there is a 50% chance that the new door you pick is correct. 50% is higher than 33% so I would have to say I would pick a new door.
J.S.
Task 1:
I would start with the first door because there is a 33% chance of picking the right door. I would keep the same door because there is another 33% for the other two doors would be right. The door is more likely to be wrong because it’s out of three doors.
N.E
Task 1: I would choose another door because there is a 66% chance that the open door is wrong and a 33% chance that it is right. If it is wrong then you have a 50% chance that the others doors are right so, since 50% is higher than 33% I would choose another door.
J.M.
Task 1.If another door was opened, I would probably keep the one that I chose because each door has the same chance of being the correct door. There is a 33% chance that you choose the right door but then you have a 50% chance of getting the right door if the door you chose is the wrong door.
EMK
I predict that switching the doors would be the best bet. Because when all doors are unopened, you have a 33.3% chance of choosing the door with the prize. When one door without the prize is revealed then you have a 66.6% chance of choosing the right door. I think that you should switch doors because then you would have a greater probability of getting the prize...?
S.S.
My prediction was correct. You do win more often when you switch then when you do not switch. Instead of winning 50% of the time, i won 60% of the time. But my prediction for Games stayed was exactly right, i won 10 of 30 games when i stayed.
JB POW
I was wrong, most of the time I switched doors I got the right door, and when I stayed with the same door I got the wrong doors. Out of 82 games I won 24 times switching doors and only won13 times staying with the same door. My first prediction was wrong, its better to switch doors.
POW MP Post 2
My prediction was incorrect. Out of fifty games on stayed, I only won 15 of them, giving me a percent of 30. With the switching you actually have about a 60 percent chance of getting the correct door. With keeping the door you actually have about a 33 percent chance of having chosen the right door.
POW EK 2nd post
My prediction was wrong. Out of 105 games each, 36 where I stayed won and 62 where I switched won. The percents are 34.29 percent and 59.05 percent. When looking at these percents, it’s basically a 66-33 chance. The door that is opened has its 33 percent chance added to the other door. This is theoretical, its technically a 50-50 chance since thers only two doors to chose from.
JW
My prediction was correct, somewhat.
When you first start the game, you have a 33.3 chance of winning the new car, because it can be behind any of the doors, so pick which ever you want. Then, when the other door opens, you now have a 66.6 percent chance if you switch doors, because, If you keep the same door, you only have a 30 percent chance of getting the new car.
Another way to think about it is, when you first start playing the game, you have a 2/3 chance of losing the game. By switching, you get the winning door on the 2 out of 3 times that you were incorrect on the first guess. By switching, you turn the odds in your favor.
Here is the probability of winning on my games, when I switched
Games switched: 50
Games switched and won: 33
Experimental probability to win: 66.00%
Here is the probability when you don’t switch:
Games stayed: 50
Games stayed and won: 15
Experimental probability: 33.00%
This shows that you have a way better chance of winning when you switch cards instead of staying.
Different example: flipping a coin. If you were to flip a coin 100 times, you would expect to get 50 heads. What would you think though, if you didn’t get that number? This is what happens while playing this game. In this case, they are 95% correct, which means 95% of the time; the outcome should fall within the percent it should fall into. In other words, only 1 time out of 20 trials should the outcome NOT fall within the range it should.
FUN FACT! When you are in a game show, and they do this, people usually play to their emotions instead of using simple math to increase their probability. That’s one of the reasons people will stay with the door, even if they know they have a better chance, because they feel like the game show host is trying to make them switch.
Ss.
as soon as you pick a door, a door opens and that door has a zero % chance. now you have two doors to pick from. Now you have a 50% chance. If you do not switch doors you limit yourself to 33% chance. Though it seems you have a 50% chacne now, it is still a 33% chance. It is better to switch doors as you have a higher percent chance.
D.E.
In my opinion, after you pick one of the doors, the host reveals a door. Then it is your door and the host’s door. In my opinion you switch because you have a better chance of getting it in the second round because one door is eliminated, and there are two doors left. Now, the hard part is to switch or stay with the door. There is a half of a chance to get the one with the prize and half a chance to get the pig or whatever the heck it is. I predict that if you switch doors you will have a 66% chance because the door that is not opened or not chosen has more of a chance to be containing a prize. I don’t really know how to explain it. When you choose your door it is chosen with a 33% of being correct. When the host opens the door, the third door has a 50% chance of being correct when the door you picked still has a 33% chance.
My prediction was wrong. There is a better chance of switching the doors. I played the game in two different rounds. The first round, I played the game only 10 times: 5 for sticking with the first door and 5 for switching doors. I won the game 5 times. 4 when I switched doors and 1 when I stayed with the first door. My experimental probability was 50%. I went back wondering if that was enough information. I played the game for a second round. This time, I played the game 20 times. 10 times where when I switched doors and the other 10 times was when I stuck with the first door. I won the game 10 times. 7 when I switched doors and 3 when I stuck with the original door. The experimental probability was 50% again. There is a 2/3 probability or chance that switching doors will win and a 1/3 probability or chance that changing doors will win. If you pick the door with the car behind it, you have a 1/3 chance of wining because if you switch, then you lose. If you pick a door with a pig behind it, you have a 2/3 chance of switching because then you would win the car.
AH
When you choose your door, you choose it with 33% chance of it being correct. When they show you a different door that is incorrect, the third door now has 50% chance of being correct while technically the original door still has 33% chance.
POW MP Post 2
My prediction was incorrect. Out of fifty games on stayed, I only won 15 of them, giving me a percent of 30. With the switching you actually have about a 66 percent chance of getting the correct door, theoretically, and when you try the original plan I made of keeping the same door you have a theoretical probability. Technically, you should have the same chance theoretically because you’re choosing between 2 doors, so technically its fifty-fifty chance but oddly enough, the experimental results differ. Of course it is possible for experimental results to differ greatly from theoretical results.
JB POW
My prediction was wrong, as you’re choosing doors the wrong one opens up changing your chances of getting the right one from 33 percent to 50-50. So your best chance at getting the right answer is to switch the door you originally chose.
JW
My prediction was correct, somewhat.
When you first start the game, you have a 33.3 chance of winning the new car, because it can be behind any of the doors, so pick which ever you want. Then, when the other door opens, you now have a 66.6 percent chance if you switch doors, because, If you keep the same door, you only have a 30 percent chance of getting the new car.
Another way to think about it is, when you first start playing the game, you have a 2/3 chance of losing the game. By switching, you get the winning door on the 2 out of 3 times that you were incorrect on the first guess. By switching, you turn the odds in your favor.
Here is the probability of winning on my games, when I switched
Games switched: 50
Games switched and won: 33
Experimental probability to win: 66.00%
Here is the probability when you don’t switch:
Games stayed: 50
Games stayed and won: 15
Experimental probability: 33.00%
This shows that you have a way better chance of winning when you switch cards instead of staying.
Different example: flipping a coin. If you were to flip a coin 100 times, you would expect to get 50 heads. What would you think though, if you didn’t get that number? This is what happens while playing this game. In this case, they are 95% correct, which means 95% of the time; the outcome should fall within the percent it should fall into. In other words, only 1 time out of 20 trials should the outcome NOT fall within the range it should.
FUN FACT! When you are in a game show, and they do this, people usually play to their emotions instead of using simple math to increase their probability. That’s one of the reasons people will stay with the door, even if they know they have a better chance, because they feel like the game show host is trying to make them switch.
JB POW
My prediction was wrong, as you’re choosing doors the wrong one opens up changing your chances of getting the right one from 33 percent to 50-50. So your best chance at getting the right answer is to switch the door you originally chose.
JML
Number of Games stayed: 55
Number of Games won: 16
Percent: 29%
Number of Games switched: 55
Number of Games won: 38
Percent: 69%
My prediction was correct.
When you first choose a door you have 1/3 of a chance of picking the car. So, you have 33.3% of choosing the prize on the first round. The 2nd round you have a 50% of getting the door right, but a higher chance of getting the prize if you switch!
O.S.
If you pick a door and it opens another door, then it will change on the probability of what you will choose. If you choose and pick the same door that you picked at first, then you have a 1/3 chance of getting it right like how you started. However, if you pick and choose a different door, then you now have a ½ chance of getting the right door.
EMK
When you first pick a door you have a 33.3% chance of getting it right and a 66.6% chance of getting it wrong. When the next door is revealed there is now 50% chance of getting the right door. So. It shouldn’t matter what you do. But. For some reason I think it would be better to switch. I am not sure though
Games stayed: 25
#of games won: 11
Experimental probability of winning: 44%
Games switched: 25
#of games won: 16
Experimental probability of winning: 61%
POW N.H.
I was right in my assumption that switching doors is the way to go because the experimental probability is higher and the percentages are higher. Also because picking your same door instead of switching gives almost a 50% chance of losing or less. If you chose a separate door than you will have a 1/3 chance of getting something other than the prize. If you don’t than you have a 2/3 have getting the prize.
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