Fun with Probability

[FrumiousBandersnatch]

Doesn't matter.

If it's a fair coin... (the question doesn't specify).

 

[crjmurray]

If it's a fair coin... (the question doesn't specify).

Well if the twist is that it is a double headed coin then that's pretty lame

 

[crjmurray]

Let's do another more simple scenario.

In our scenario, we have David. David has a sibling. We do not know if the sibling is older or younger. What is the probability that the sibling is a male?

 

[keith99]

That's a simple and elegant way to look at it, morningstar. I think that would have convinced me immediately when I first heard this problem. The one that actually reached me first was (as previously mentioned) extending it to 1000 doors and one car, where you pick a door (1/1000) and all the other doors, except the one with the car, are opened. Guess I'm a sucker for reductio ad absurdum

Now I understand the 1000 door comments. That makes it intuitively obvious.

I think one thing with just the 3 doors and one example is that on intuitively thinks of door 3 being opened. That is not the case, it is a door with a goat.

Technically your example is off. 1 time out of 1000 you pick the car and that time it is all the other doors except one that get opened. Which for me just reinforces the intuitive aspects matching the reality.

 

[keith99]

Well if the twist is that it is a double headed coin then that's pretty lame

No the twist is more subtle. The fools answer is tails because it has to even out.

The better answer is it does not matter.

Dr. Pournelle's answer is if it is an honest coin and there is nothing strange going on then it does not matter. But if it is not an honest coin you should bet heads. Thus you should bet heads. Note this also holds for any situation where the coin is not flat out rigged but not really even. If for example the coin has a very strange balance and it comes heads 90% of the time then the odds of 8 heads in a row is much higher than 8 tails.

 

[Subduction Zone]

That's a simple and elegant way to look at it, morningstar. I think that would have convinced me immediately when I first heard this problem. The one that actually reached me first was (as previously mentioned) extending it to 1000 doors and one car, where you pick a door (1/1000) and all the other doors, except the one with the car, are opened. Guess I'm a sucker for reductio ad absurdum

To be clearer you could have said:

After you make your choice Monty opens up 998 doors, all showing a goat. Now the only unopened doors are the one you originally chose or the one that Monty did not open.

Do you switch or stay?

You are assuming, not much of an assumption really, that the last remaining door, besides yours, had the car behind it. Sadly some people cannot understand that this is the same problem, only with more doors.

 

[morningstar2651]

Here is a fun basic one: How many people must be in a room before the probability that some share a birthday becomes at least 50 percent?

Birth year doesn't matter.

 

[crjmurray]

Here is a fun basic one: How many people must be in a room before the probability that some share a birthday becomes at least 50 percent?

Birth year doesn't matter.

Without doing the math, I think it's around 20/21 if I remember correctly.

 

[morningstar2651]

Without doing the math, I think it's around 20/21 if I remember correctly.

Very close.

It's called the birthday paradox.

 

[AV1611VET]

Since we have so many experts on probability here, I thought we could play around with some puzzles. We'll start with an oldie.

"Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?"

Well? Do you stay with your original choice or do you switch? By staying or switching does one give you any better/worse odds?

Suppose you pick door #1, but you lie to the host and say door #3.

The host then opens door #1, which has a goat.

Knowing you lied, should you stick with door #3, or switch?

 

[jacks]

Love it.

 

[AV1611VET]

Love it.

Thanks!

 

[Subduction Zone]

Suppose you pick door #1, but you lie to the host and say door #3.

The host then opens door #1, which has a goat.

Knowing you lied, should you stick with door #3, or switch?

Too late.

 

[Nick665]

Suppose you pick door #1, but you lie to the host and say door #3.

The host then opens door #1, which has a goat.

Knowing you lied, should you stick with door #3, or switch?

You just won a goat .

 

[AV1611VET]

You just won a goat .

How?

By sticking or switching?

 

[Subduction Zone]

How?

By sticking or switching?

Someone that understood Boolean Algebra would understand.

 

[Architeuthus]

I have to post this video now (if the new software will let me).

GUILDENSTERN: Heads. He keeps flipping the coin. Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads …He throws it up again, and Rosencrantz catches it, looks at the coin and throws it back to Guildenstern. Heads. Rosencrantz gets a coin out of his purse and flips it, covering it with has hand. Bet. Heads, I win. Rosencrantz looks at the coin, says nothing, and throws it to Guildenstern, who covers it with his hand. Again. Guildenstern looks at the coin. Heads. Rosencrantz gets out another coin.
The two continue on horseback, with Guildenstern flipping a coin as he rides. Heads … Heads … Heads … Heads … he drops the coin, and it rolls down the hill. It comes up heads.
GUILDENSTERN: It must be indicative of something besides the redistribution of wealth. He flips a coin to Rosencrantz, who looks at it.
ROSENCRANTZ: Heads.
GUILDENSTERN: A weaker man might be moved to reexamine his faith. If nothing else, at least in the law of probability. He flips another coin to Rosencrantz.
ROSENCRANTZ: Heads.
GUILDENSTERN: Consider. One. Probability is a factor which operates within natural forces. Two. Probability is not operating as a factor. Three. We are now held within un- sub- or super-natural forces. Discuss.

 

[Nick665]

How?

By sticking or switching?

No, by deserving it.

 

[[serious]]

I have to post this video now (if the new software will let me).

GUILDENSTERN: Heads. He keeps flipping the coin. Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads … Heads …He throws it up again, and Rosencrantz catches it, looks at the coin and throws it back to Guildenstern. Heads. Rosencrantz gets a coin out of his purse and flips it, covering it with has hand. Bet. Heads, I win. Rosencrantz looks at the coin, says nothing, and throws it to Guildenstern, who covers it with his hand. Again. Guildenstern looks at the coin. Heads. Rosencrantz gets out another coin.
The two continue on horseback, with Guildenstern flipping a coin as he rides. Heads … Heads … Heads … Heads … he drops the coin, and it rolls down the hill. It comes up heads.
GUILDENSTERN: It must be indicative of something besides the redistribution of wealth. He flips a coin to Rosencrantz, who looks at it.
ROSENCRANTZ: Heads.
GUILDENSTERN: A weaker man might be moved to reexamine his faith. If nothing else, at least in the law of probability. He flips another coin to Rosencrantz.
ROSENCRANTZ: Heads.
GUILDENSTERN: Consider. One. Probability is a factor which operates within natural forces. Two. Probability is not operating as a factor. Three. We are now held within un- sub- or super-natural forces. Discuss.

I suspect that there are more than 262144 people who have flipped coins, thus an isolated series of 18 heads is not unexpected.

 

[[serious]]

Before the door was opened, the possibilities were:

Door 1.       Door 2.       Door 3
Car.         Goat 1.       Goat 2
Car.         Goat 2.       Goat 1
Goat 1.       Car.         Goat 2
Goat 1.       Goat 2.        Car
Goat 2.       Car.          Goat 1
Goat 2.       Goat 1.        Car

After the door is opened, possibilities 4 and 6 are ruled out. So the remaining possibilities are:

Door 1.        Door 2.        Door 3
Car.          Goat 1.        Goat 2
Car.          Goat. 2.       Goat 1
Goat 1.        Car.              Goat 2
Goat 2.        Car.              Goat 1

Therefore the conditional probability of a car being behind door 1 is 1/2

The problem with that logic is that those options 4 and 6 aren't removed, they are actively avoided, converted into another option if you will. You are seeing 4 remaining options and assuming they are all equally probable, but they aren't. Let's look at it this way instead.

You have a 33% chance of picking a car or either goat. The host has a 100% chance of choosing a goat because he knows where the goats are and is actively choosing them.

So let's look at those options again:

Yours       Remaining   Hosts      
Car.         Goat 1.       Goat 2
Car.         Goat 2.       Goat 1
Goat 1.       Car.         Goat 2
Goat 1.       Goat 2.        Car
Goat 2.       Car.          Goat 1
Goat 2.       Goat 1.        Car

Now, it isn't that 4 and 6 just don't happen, but instead they are replaced by the host choosing the goat instead.

Yours       Remaining   Hosts       
Car.         Goat 1.       Goat 2
Car.         Goat 2.       Goat 1
Goat 1.       Car.         Goat 2
Goat 1.       Car          Goat 2.       
Goat 2.       Car.         Goat 1
Goat 2.       Car          Goat 1.