What is the probability of ?

All_Is_Number · 2008-10-07 15:24:04

I've come across a probability problem for which I've seen multiple different answers proposed, but none with a satisfactory explanation of the logic leading to the proposed result. I'll give the problem in general terms, and then with two examples.

Please offer sufficient justification for your answer. I hope you have better luck than I've had.

Generally:

Given:

Each element in a sample space S has two positions, and each position has a primary characteristic, X or Y, and a secondary characteristic, a or b.

S = {(Xa,Xa), (Xa,Xb), (Xb,Xa), (Xb,Xb), (Xa,Ya), (Xa,Yb), (Xb,Ya), (Xb,Yb), (Ya,Xa), (Ya,Xb), (Yb,Xa), (Yb,Xb), (Ya,Ya), (Ya,Yb), (Yb,Ya), (Yb,Yb)};

XX ⊊ S; XX = {(Xa,Xa), (Xa,Xb), (Xb,Xa), (Xb,Xb)};

XY ⊊ S; XY = {(Xa,Ya), (Xa,Yb), (Xb,Ya), (Xb,Yb)};

YX ⊊ S; YX = {(Ya,Xa), (Ya,Xb), (Yb,Xa), (Yb,Xb)};

YY ⊊ S; YY = {(Ya,Ya), (Ya,Yb), (Yb,Ya), (Yb,Yb)};

{XX, XY, YX, YY} ≡ S.

X ≡ (¬Y) and a ≡ (¬b).

P(S) = 1.

P(X) = P(Y) = ½.

It is not necessarily known if P(a) for the second position is dependent or independent of the secondary characteristic of the first position.

It is not necessarily known if P(a | X) = P(a | Y).

Find the probability that an event is an element of YY, given that one of its positions is Ya.

Example 1:

Two (fair) pennies are flipped, sequentially. What is the probability that both pennies land heads up if it is known that one of them landed heads up and was minted in 1980. (We do not know if it was the first or the second penny that landed heads up.)

Example 2.

A family is known to have two children. What is the probability that both children are girls if it is known that one of the children is a girl with the name Morgan. Assume boys and girls are equally likely to be born and survive. (Note that Morgan can be given to boys or girls and can also be a first, middle or last name.)

Last edited by All_Is_Number (2008-10-08 00:04:38)

mathsyperson · 2008-10-07 19:44:01

This is an ambiguous question that could be equivalent to either one of these:

1. Two pennies are flipped. What is the probability that they both come up heads, given that one of them is a head?
2. Two coins (1p and 2p) are flipped. What is the probability that they come up heads, given that the 1p is a head?

For 1, there are three possibilities (HH, HT, TH) so the answer is 1/3.
For 2, there are only two possibilities (HH, HT) so the answer is 1/2.

So the best answer I can think of to your question would be 1/3P(x) + 1/2(1-P(x)), where P(x) is an additional probability. In the pennies case, it would be the probability that a random coin was minted in 1980. Taking that as 1/50, for example, would give the final answer to be 149/300.

Since additional information is required, though, that won't work on the abstracted version.

All_Is_Number · 2008-10-07 23:42:21

mathsyperson wrote:

This is an ambiguous question that could be equivalent to either one of these:
1. Two pennies are flipped. What is the probability that they both come up heads, given that one of them is a head?
2. Two coins (1p and 2p) are flipped. What is the probability that they come up heads, given that the 1p is a head?

Option 1. We don't know which penny came up heads, only that at least one of the two did. The one of the two that did come up heads was minted in 1980.

I know that any answers are going to be in terms of P(a).

I'm not sure I follow how you arrived at your answer.

Last edited by All_Is_Number (2008-10-07 23:45:22)

mathsyperson · 2008-10-08 07:08:36

My point was that the answer changes depending on whether the additional information is useful or not.

If both coins you tossed were minted in 1980, then the information that the head came up on a 1980 coin is unhelpful and the question is of form 1.

If the other coin was minted in some other year, then the coins can be told apart and the question is of form 2.

The answer I gave was basically
P(question is in form 1) x P(other coin is a head, given that the question is form 1)
+
P(question is in form 2) x P(other coin is a head, given that the question is form 2)

Some of the abstract stuff you posted is going over my head though, so it's possible I'm misinterpreting your question.

All_Is_Number · 2008-10-10 04:05:28

mathsyperson wrote:

My point was that the answer changes depending on whether the additional information is useful or not.
If both coins you tossed were minted in 1980, then the information that the head came up on a 1980 coin is unhelpful and the question is of form 1.
If the other coin was minted in some other year, then the coins can be told apart and the question is of form 2.
The answer I gave was basically
P(question is in form 1) x P(other coin is a head, given that the question is form 1)
+
P(question is in form 2) x P(other coin is a head, given that the question is form 2)
Some of the abstract stuff you posted is going over my head though, so it's possible I'm misinterpreting your question.

Well, the question was definitely intended to be of form one (despite any unintentional ambiguity on my part).

I solved the coin version of the problem. It was simpler, since we can assume that the mint date is not dependent on heads, tails or position (i.e. flip 1 or 2).

Letting P(a) be the probability of a penny being minted in 1980, and applying the formula for conditional probability, P(Both pennies land heads | At least one penny landed heads and was minted in 1980) =

[(¼)P(a)² + (¼)(P(a)-P(a)²) + (¼)(P(a)-P(a)²)]
÷ [(¼)P(a)² + (¼)(P(a)-P(a)²) + (¼)(P(a)-P(a)²) + (¼)P(a)² + (¼)(P(a)-P(a)²) + (¼)P(a)² + (¼)(P(a)-P(a)²)]

= (P(a)-2)/((P(a)-4).

As P(a) approaches zero, this expression approaches 1/2.

I have a nice chart illustrating where each of the above expressions came from, but was unable to upload it.

Something I found very interesting: your method resulted in a very close approximation of the exact answer (in terms of P(a)). The maximum difference between the two expressions is less than 0.012, at P(a)≈0.5359. For more realistic values of P(a), say 0.05 or less, the difference is 0.002 or less.

Last edited by All_Is_Number (2008-10-10 04:07:17)

Math Is Fun Forum

#1 2008-10-07 15:24:04

What is the probability of ?

#2 2008-10-07 19:44:01

Re: What is the probability of ?

#3 2008-10-07 23:42:21

Re: What is the probability of ?

#4 2008-10-08 07:08:36

Re: What is the probability of ?

#5 2008-10-10 04:05:28

Re: What is the probability of ?

Board footer