A mad scientist

Here's a little snack-sized puzzle. I saw this on a science magazine some years ago and modified it a little.

You're in a lab of a mad scientist. You see two identical containers on the table but you can't see their contents. When you look at the scientist, you notice that he's holding a fresh human brain in his hands. "One of those containers", he says, "has either a liver or a brain in it. The other one is empty." He doesn't tell you which is which.

Then he puts the brain he's holding into one container and shuffles them until you have no idea which one it was. He proceeds by picking one of the containers and pulling a brain out of it. He puts the brain on the table and points the container he had picked. "With what probability does that jar have another brain in it?"

0.125. This was my guess after about 15 seconds of thought in the "space landing" thread. I was wrong. After about a minute of thinking, I think it stands a good shot at this one, or at least, it's a good wrong answer --- if that exists.

No! no! no! I changed my mind, I think it's 1/3! Yeah. Am I right? Am I right?

Actually, it might be 1/5.

I give up.

Can the containers hold two objects? i.e. Could he have initially put the brain he was holding into the container already holding a brain or liver?

Originally posted by Fat Lady
Can the containers hold two objects? i.e. Could he have initially put the brain he was holding into the container already holding a brain or liver?

Yes. If the containers could hold only one object each, the jar would be empty after taking the brain out. Thus, the probability would be zero and the puzzle would be pointless.

Originally posted by Keltamaksa
Yes. If the containers could hold only one object each, the jar would be empty after taking the brain out. Thus, the probability would be zero and the puzzle would be pointless.

I think it's 9/16. The possible contents of the containers after placing the brain in one of them are as follows:

1) 2 brains, empty
2) brain, brain
3) brain + liver, empty
4) brain, liver

The possible outcomes of drawing at random (with probability weightings) are as follows:

1) brain (P=1/4), brain (1/4), nothing (1/2)
2) brain (1/2), brain (1/2)
3) brain (1/4), liver (1/4), empty (1/2)
4) brain (1/2), liver (1/2)

The sum of these probabilities (without normalizing) 4. Of these, only:

1) brain (1/4), brain (1/4);
2) brain (1/2), brain (1/2);
3) brain (1/4); and
4) brain (1/2)

could have happened given that a brain was chosen. The sum of these probabilities (without normalizing) of these events is 9/4. So the probability that a second brain is in the jar is (9/4)/4 = 9/16.

A man has exactly two children. At least one of them is a boy.
What is the probability that both his children are boys?
----------------------------------------------------------------------
A woman has exactly two children. The older of the two is a boy.
What is the probability that both her children are boys?

A box contains exactly two coins, either two Silver or one Gold and one Silver. A coin is chosen at random. It is Silver.
What is the probability that the other coin is also Silver?

My theory is the following:

chance that the jar he picked had 2 items = 50%
chance that the second item is a brain = 50%

so chance of the two brains being in one jar = 25%

Actually, that sounds pretty good too. I want to switch to 25%. What's the answer --- I'm dying here!

Originally posted by Keltamaksa
Here's a little snack-sized puzzle. I saw this on a science magazine some years ago and modified it a little.

You're in a lab of a mad scientist. You see two identical containers on the table but you can't see their contents. When you look at the scientist, you notice that he's holding a fresh human brain in his hands. "One of those containers", he say ...[text shortened]... ainer he had picked. "With what probability does that jar have another brain in it?"

Let the two containers be named 1 and 2.

"One of those containers", he says, "has either a liver or a brain in it. The other one is empty."

Two possibilities; 1 is empty or 2 is empty.

Two possibilities; the non empty one has a brain or the non empty one has a liver. 2x2 = 4 possibilities. Let these four possibilites be named 1L, 1B, 2L, 2B.

Then he puts the brain he's holding into one container and shuffles them until you have no idea which one it was.

He either put the brain into 1 or 2. Now we have eight possibilities. Let them be named 1L1B, 1L2B, 1B1B, 1B2B, 2L1B, 2L2B, 2B1B and 2B2B.

He proceeds by picking one of the containers and pulling a brain out of it. He puts the brain on the table and points the container he had picked. "With what probability does that jar have another brain in it?"

Only in the cases 1B1B and 2B2B will the jar have another brain in it. That's 2 of the 8 possibilities. Thus the answer is P = 0.25.

Originally posted by PBE6
I think it's 9/16. The possible contents of the containers after placing the brain in one of them are as follows:

1) 2 brains, empty
2) brain, brain
3) brain + liver, empty
4) brain, liver

The possible outcomes of drawing at random (with probability weightings) are as follows:

1) brain (P=1/4), brain (1/4), nothing (1/2)
2) brain (1/2), brain (1/ ...[text shortened]... f these events is 9/4. So the probability that a second brain is in the jar is (9/4)/4 = 9/16.

I can't find your mistake. Can anyone else? Or can they find my mistake in case this solution is correct?

EDIT - I think the mistake is that you assume knowledge you don't have; that is, you don't know that the Mad Scientist reached into the jars randomly and pulled out a random organ. This assumption of yours was necessary for this step which was therefore flawed:

The possible outcomes of drawing at random (with probability weightings) are as follows:

Originally posted by AThousandYoung
I can't find your mistake. Can anyone else? Or can they find my mistake in case this solution is correct?

EDIT - I think the mistake is that you assume knowledge you don't have; that is, you don't know that the Mad Scientist reached into the jars randomly and pulled out a random organ. This assumption of yours was necessary for this step which was ...[text shortened]... ]The possible outcomes of drawing at random (with probability weightings) are as follows:[/b]

I think you have to assume the game is fair (i.e. the scientist picks randomly), otherwise you have to make several additional assumptions which are tenuous at best. Even if he doesn't, but you don't know about his scheme, you can still answer the question truthfully and correctly.

However, I think my original solution was wrong. This is a case of conditional probabilty, so we use the formula P(A given B) = P(A and B)/P(B), where A = 2 brains in one jar, and B = a brain was chosen. The possible outcomes and weightings (normalized this time) are as follows:

1) 2 brains, empty

brain (P=1/16), brain (1/16), empty (1/8)

2) brain, brain

brain (1/8), brain (1/8)

3) brain + liver, empty

brain (1/16), liver (1/16), empty

4) brain, liver

brain (1/8), liver (1/8)

The probability that there are 2 brains in the jar and a brain was picked is simply 2*(1/16) = 1/8 from (1) above. The probability that a brain was picked is (1/16) + (1/16) + (1/8) + (1/8) + (1/16) + (1/8) = 9/16. Now using the formula for conditional probability, we get:

P(A given B) = P(A and B)/P(B) = (1/8)/(9/16) = 2/9

My original solution did not take into account the chance of the brain being from the 2 brain jar. My brain must have been in a jar at the time. 🙄

Originally posted by ThudanBlunder
A man has exactly two children. At least one of them is a boy.
What is the probability that both his children are boys?
----------------------------------------------------------------------
A woman has exactly two children. The older of the two is a boy.
What is the probability that both her children are boys?

PART A

We assume the probability of having a child of either sex is 1/2. The possible outcomes of the man's shagging are:

1) BB
2) BG
3) GB
4) GG

All of the above have a probability of 1/4. (1), (2) and (3) have at least one boy, but only (1) has two boys. Therefore, the probability that the man's other child is also a boy is (1/4)/(3/4) = 1/3.

PART B

The possible outcomes of the woman's shagging, given that the older child is a boy, are:

1) BB
2) GB

Both of the above have a probability of 1/2. Both have an older boy, but only (1) has two boys. Therefore the probability that the woman's other child is also a boy is (1/2)/1 = 1/2.

Originally posted by ThudanBlunder
A box contains exactly two coins, either two Silver or one Gold and one Silver. A coin is chosen at random. It is Silver.
What is the probability that the other coin is also Silver?

The possibile outcomes are:

1) Silver coin, silver side 1
2) Silver coin, silver side 2
3) Mixed coin, silver side
4) Mixed coin, gold side

Each of the above occur with probability 1/4. (1), (2) and (3) all show a silver side, but of those (1) and (2) have both sides silver. Therefore the probability that the other side of the coin is also silver is (2/4)/(3/4) = 2/3.