# What is the probability that this person is female?

There is a person behind a curtain - I do not know whether the person is female or male.

I know the person has long hair, and that 90% of all people with long hair are female

I know the person has a rare blood type AX3, and that 80% of all people with this blood type are female.

What is the probability the person is female?

NOTE: this original formulation has been expanded with two further assumptions: 1. Blood type and hair length are independent 2. The ratio male:female in the population at large is 50:50

(The specific scenario here is not so relevant - rather, I have an urgent project that requires I get my mind around the correct approach for answering this. My gut feel is that it's a question of simple probability, with a simple definitive answer, rather than something with multiple debatable answers according to different statistical theories.)

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There are not multiple theories of probability, but it is notoriously true that people have difficulties thinking correctly about probabilities. (Augustus DeMorgan, a good mathematician, gave up the study of probability due to its difficulties.) Don't look at debates: look for appeals to principles of probability (such as the Kolmogorov axioms). Don't let this be resolved democratically: your question is attracting many ill-conceived answers which, even if some of them happen to agree, are merely collectively wrong. @Michael C gives good guidance; my reply tries to show you why he's right. –  whuber Jun 22 '12 at 3:46
@Whuber, if independence is assumed, would you agree that 0.97297 is the correct answer? (I believe that the answer might be anywhere between 0% and 100% without this assumption - your diagrams show this nicely). –  ProbablyWrong Jun 22 '12 at 4:53
Independence of what, precisely? Are you suggesting that female and male hairstyles are the same? As you say in your question, this particular scenario involving gender/hair/blood type may not be relevant: that tells me you seek to understand how to solve problems like this in general. To do that you will need to know which assumptions imply which conclusions. Thus you need to focus very carefully on the assumptions you are willing to make and determine exactly how much they allow you to conclude. –  whuber Jun 22 '12 at 13:07
The kind of independence to explore concerns the combination of all three characteristics. E.g., if AX3 is a marker for a syndrome that includes baldness in females (but not in males), then any long-haired person with AX3 is necessarily male, making the probability of being female 0%, not 97.3%. I hope this makes it obvious that anybody producing a definite answer to this question must be making additional assumptions, even if they do not explicitly acknowledge them. The truly useful answers, IMHO, would be those that show directly how different assumptions lead to different results. –  whuber Jun 22 '12 at 14:31
You're missing the probability that a female doesn't have long hair. That's a critical measure. –  Daniel R Hicks Jul 3 '12 at 19:54

Many people find it helpful to think in terms of a "population," subgroups within it, and proportions (rather than probabilities). This lends itself to visual reasoning.

I will explain the figures in detail, but the intention is that a quick comparison of the two figures should immediately and convincingly indicate how and why no specific answer to the question can be given. A slightly longer examination will suggest what additional information would be useful for determining an answer or at least obtaining bounds on the answers.

### Legend

Cross-hatching: female / Solid background: male.

Top: long-haired / Bottom: short-haired.

Right (and colored): AX3 / Left (uncolored): non-AX3.

### Data

Top cross-hatching is 90% of the top rectangle ("90% of all people with long hair are female").

Total cross-hatching in the right colored rectangle is 80% of that rectangle ("80% of all people with this blood type are female.")

### Explanation

This diagram shows schematically how the population (of all females and non-females under consideration) can simultaneously be partitioned into females/non-females, AX3/non-AX3, and long haired/non-long haired ("short"). It uses area, at least approximately, to represent proportions (there's some exaggeration to make the picture clearer).

It is evident that these three binary classifications create eight possible groups. Each group appears here.

The information given states that the upper cross-hatched rectangle (long-haired females) comprises 90% of the upper rectangle (all long-haired people). It also states that the combined cross-hatched parts of the colored rectangles (long-haired females with AX3 and short-haired females with AX3) comprise 80% of the colored region at the right (all people with AX3). We are told that someone lies in the upper right corner (arrow): long-haired people with AX3. What proportion of this rectangle is cross-hatched (female)?

I have also (implicitly) assumed that blood type and hair length are independent: the proportion of the upper rectangle (long hair) that is colored (AX3) equals the proportion of the lower rectangle (short hair) that is colored (AX3). That's what independence means. It is a fair and natural assumption to make when addressing such questions like this, but of course it needs to be stated.

The position of the upper cross-hatched rectangle (long-haired females)is unknown. We can imagine sliding the top cross-hatched rectangle side-to-side and sliding the bottom cross-hatched rectangle side-to-side and possibly changing its width. If we do this so that 80% of the colored rectangle remains cross-hatched, such an alteration will change none of the stated information, yet it can alter the proportion of females in the upper right rectangle. Evidently the proportion could be anywhere between 0% and 100% and still be consistent with the information given, as in this image:

One strength of this method is it establishes the existence of multiple answers to the question. One could translate all this algebraically and, by means of stipulating probabilities, offer specific situations as possible examples, but then the question would arise whether such examples are really consistent with the data. For instance, if someone were to suggest that perhaps 50% of long-haired people are AX3, at the outset it is not evident that this is even possible given all the information available. These (Venn) diagrams of the population and its subgroups make such things clear.

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Whuber, assuming that blood type and hair length are independent, then surely the portion of long haired women with type AX3 should be the same as the portion of short haired women with AX3? I.e. you don't have flexibility to shift rectangles in the way you propose... If we assume also that men and women are 50:50 in the whole population, doesn't that give us enough info to solve this question with a single indisputable answer? –  ProbablyWrong Jun 21 '12 at 6:40
@whuber +1 very nice. –  Michael Chernick Jun 21 '12 at 10:42
ProbablyWrong, take a close look at the question in your comment: because it deals with women, it is making an additional assumption about independence conditional on gender. The assumption of (unconditional) independence of hair and blood type does not mention gender at all, so to understand what it means, erase the cross-hatching from the figures. This, I hope, indicates why we have the flexibility to situate the cross-hatching wherever we like within the upper and lower rectangles. –  whuber Jun 21 '12 at 14:46
@whuber, I like this. However, I have 2 questions / clarifications: 1. the figures seem to assume population proportions for long vs short hair (about 6:4) & ~AX3 vs AX3 (about 85:15), but this is not mentioned in the original question nor discussed in your explanations of the figures. I suspect the pop proportions are not relevant. Am I right / could you clarify that in the explanations? 2. I think this situation is ultimately working w/ the same phenomenon as Simpson's Paradox, only framed differently (coming at the issue from the other direction, as it were). Is that a fair assessment? –  gung Jun 27 '12 at 16:25
@gung, thank you for making those clarifications. The figures of course must represent some proportions in order to work at all, but any proportions not specifically pinned down in the problem statement are free to vary. (I did construct the figure so that about 50% of the population appears female, anticipating a later edit in which this was assumed.) The idea of applying this graphical representation to understanding Simpson's Paradox is intriguing; I think it has merit. –  whuber Jun 27 '12 at 16:34

This is a question of conditional probability. You know that the person has long hair and blood type Ax3 . Let$$\ \ \ \ \ A =\{\text{'The person has long hair'}\}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ B = \{\text{'The person has blood type Ax3'}\} \\ C =\{\text{'The person is female'}\}.$$
So you seek $P(C|A\ \text{and}\ B)$. You know that $P(C|A)=0.9$ and $P(C|B)=0.8$.
Is that enough to calculate $P(C|A\ \text{and}\ B)$? Suppose $P(A\ \text{and}\ B\ \text{and}\ C)=0.7$. Then $$P(C|A\ \text{and}\ B)=P(A\ \text{and}\ B\ \text{and}\ C)/ P(A\ \text{and}\ B)=0.7/P(A\ \text{and}\ B).$$ Suppose $P(A\ \text{and}\ B)=0.8$. Then, by the above, $P(C|A\ \text{and}\ B)=0.875$. On the other hand if $P(A\ \text{and}\ B)=0.9$ we would then have $P(C|A\ \text{and}\ B)$=0.78.

Now both are possible when $P(C|A)=0.9$ and $P(C|B)=0.8$. So we can't tell for sure what $P(C|A\ \text{and}\ B)$ is.

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Hi Michael, If I read you correctly, you're saying the question as posed can't be answered, is that right? Or to put it another way, you'd need more information to answer this question? 1. Let's assume that the rare blood type in my original question doesn't have any impact on a person's desire or ability to grow their hair long. Can the question now be answered? 2. Would you agree that the answer must be GREATER than 0.9? (Because you have a second piece of independent information - blood type - that reinforces the hypothesis that the person is a female) –  ProbablyWrong Jun 21 '12 at 3:30
If $P(A\text{ and }B)$ is independant, then $P(A\text{ and }B)=P(A)P(B)$ and you'll need to specify what fraction of persons have long hair, i.e., $P(A)$ and what fraction of persons have blood type Ax3, i.e., $P(B)$. Also, you can't say that the answer must be greather than 0.9, which is equivalent to stating that $P(C|A\text{ and }B)>0.9$ (I really don't see why). –  Néstor Jun 21 '12 at 7:39
Thank you Nestor. –  Michael Chernick Jun 21 '12 at 10:00
@ProbablyWrong. Yes the problem as initially stated has insufficient information for a unique answer. –  Michael Chernick Jun 21 '12 at 10:04
Why does $$P(C|A\ \text{and}\ B)=P(A\ \text{and}\ B\ \text{and}\ C)\times P(A\ \text{and}\ B)??$$ I thought that $$P(C|A\cap B)=\frac{P(C \cap (A \cap B))}{P(A \cap B)}=\frac{P(A\cap B\cap C)}{P(A\cap B)}$$ using the definition of conditional probability. –  Dilip Sarwate Jun 21 '12 at 11:13

Fascinating discussion ! I am wondering if we specified P(A) and P(B) as well whether the ranges of P(C| A,B) will not be much narrower than the full interval [0,1], simply because of the many constraints we have.

Sticking to the notation introduced above:

A = the event that the person has long hair

B = the event that the person has blood type AX3

C = the event that person is female

P(C|A) = 0.9

P(C|B) = 0.8

P(C) = 0.5 (i.e. let's assume an equal ratio of men and women in the population at large)

it does not seem possible to assume that events A and B are conditionally independent given C ! That leads directly to a contradiction: if $P(A \wedge B | C) = P(A| C) \cdot P(B| C) = P(C| A) \frac{P(A)}{P(C)} \cdot P(C| B) \frac{P(B)}{P(C)}$

then

$P(C| A \wedge B ) = P(A \wedge B | C) \cdot \left( \frac{P(C)}{P(A \wedge B)} \right) = P(C| A) \frac{P(A)}{P(C)} \cdot P(C| B) \frac{P(B)}{P(C)} \cdot \left( \frac{P(C)}{P(A \wedge B)} \right)$

If we now assume that A and B are independent as well: $P(A \wedge B) = P(A) P(B)$ most terms cancel and we end up with

$P(C| A \wedge B ) = \frac{P(C| A) \cdot P(C| B)}{P(C)} = \frac{0.9 \cdot 0.8}{0.5} > 1$

Following up on whuber's wonderful geometric representation of the problem: While it is true that generally speaking $P(C | A \wedge B)$ can assume any value in the interval $[0,1]$ the geometric constraints do narrow the range of possible values significantly for values of $P(A)$ and $P(B)$ that are not "too small". (Though we can also upper bound the marginals: $P(A)$ and $P(B)$)

Let us compute the {\bf smallest possible value} for $P(C | A \wedge B)$ under the following geometric constraints:

1. The fraction of the upper area (A TRUE) covered by the upper rectangle must be equal to $P(C|A)=0.9$

2. The sum of the areas of the two rectangles must be equal to $P(C)=0.5$

3. The sum of the fraction of the areas of the two colored rectangles (i.e. their overlap with event B) must be equal to $P(C|B)=0.8$

4. (trivial) The upper rectangle cannot be moved beyond the left boundary and should not be moved beyond its minimum overlap to the left.

5. (trivial) The lower rectangle cannot be moved beyond the right boundary and should not be moved beyond its maximum overlap to the right.

These constraints limit how freely we can slide the hashed rectangles and in turn generate lower bounds for $P(C | A \wedge B)$. The figure below (created with this R script ) shows two examples

Running through a range of possible values for P(A) and P(B) (R script) generates this graph

In conclusion, we can lower bound the conditional probability P(c|A,B) for given P(A), P(B)

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Markus, the first paragraph belongs as a separate question rather than within an answer. The subsequent material looks like a good observation but it is hard to follow without being told what $A, B,$ and $C$ represent. Please bear in mind that different users will see the answers in different sequences, according to their preferences and when the answers were last edited, so each answer has to be readable independently of the others (although of course you can link to other answers). –  whuber Jul 3 '12 at 14:34
@whuber: thanks for the useful comment ! I hope the new edits make it more readable and clear. –  Markus Loecher Jul 3 '12 at 18:36
@whuber and others: I had hoped to reignite the discussion but the thread seems to have gone inactive ? No more comments by anyone ? –  Markus Loecher Jul 8 '12 at 20:36

Make the hypotheses is that the person behind a curtain is a woman.

We area given 2 pieces of evidence, namely:

Evidence 1: We know the person has long hair (and we're told that 90% of all people with long hair are female)

Evidence 2: We know the person has a rare blood type AX3 (and we're told that 80% of all people with this blood type are female)

Given just Evidence 1, we can state that the person behind a curtain has a 0.9 probability value of being a woman (assuming 50:50 split between men and women).

Regarding the question posed earlier in the thread, namely "Would you agree that the answer must be GREATER than 0.9?", without doing any Math, I would say intuitively, the answer must be "yes" (it is GREATER than 0.9). The logic is that Evidence 2 is supporting evidence (again, assuming a 50:50 split for the number of men and women in the world). If we were told that 50% of all people with AX3 type blood were female, then Evidence 2 would be neutral and have no bearing. But since we're told that 80% of all people with this blood type are female, Evidence 2 is supporting evidence and logically should push the final probability of a woman above 0.9.

To calculate a specific probability, we can apply Bayes' rule for Evidence 1 and then use Bayesian updating to apply Evidence 2 to the new hypothesis.

Suppose:

A = the event that the person has long hair

B = the event that the person has blood type AX3

C = the event that person is female (assume 50%)

Applying Bayes rule to Evidence 1:

P(C|A) = (P(A|C) * P(C)) / P(A)

In this case, again if we assume 50:50 split between men and women:

P(A) = (0.5 * 0.9) + (0.5 * 0.1) = 0.5

So, P(C|A) = (0.9 * 0.5) / 0.5 = 0.9 (Not surprising, but it would be different if we didn't have 50:50 split between men and women)

Using Bayesian updating to apply Evidence 2 and plugging in 0.9 as the new prior probability, we have:

P(C|A AND B) = (P(B|C) * 0.9) / P(E)

Here, P(E) is the probability of Evidence 2, given the hypotheses that the person already has a 90% chance of being female.

P(E) = (0.9 * 0.8) + (0.1 * 0.2) [this is law of total probability: (P(woman)*P(AX3|woman) + P(man)*P(AX3|man)] So, P(E) = 0.74

So, P(C|A AND B) = (0.8 * 0.9) / 0.74 = 0.97297

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There are a few statements in your answer that do not make sense to me. (1) P(C|A)=0.9 by assumption. Nowhere was it said that P(C)=0.9. We assumed P(C)=0.5. (2) How did you get the result for P(E)? P(woman)=P(man)=0.5 by assumption where you write P(woman)=0.9. –  Michael Chernick Jun 21 '12 at 11:39
The value of P(C) is assumed at 0.5, which is what I've used. The value for P(E) is the probability of Evidence 2 after applying Evidence 1 (which leads to a new hypotheses that the probability that the person is female is 0.9). P(E) = (probability that the person is a woman (given Evience 1) * probability the the person has AX3 if a woman) + (probability that the person is a man (given Evience 1) * probability the the person has AX3 if a man) = (0.9 * 0.8) + (0.1 * 0.2) = 0.74 –  RandomAnswer Jun 21 '12 at 14:42
Your definition of probability of E is a bit confusing and the terms you are using to calculate it look different from what you wrote before. It really doesn't matter though. The answer is apparently correct based on Huu's nicely presented answer. –  Michael Chernick Jun 21 '12 at 14:57
@Michael Except it appears Huu made mistakes. –  whuber Jun 21 '12 at 15:12
This answer is simply wrong. There may be other errors, but this one is glaring. You state a definitive answer for P("Has Long Hair") (your P(A)), and then use that to give your final definitive answer. There simply isn't enough information to determine this, even assuming P(F) = 0.5. Your line to calculate P(A) seems to come from nowhere. Here is the correct formula using Bayes theroem: P(A) = P(A|F)P(F)/P(F|A) from which, using your stated assumptions, get to P(A) = P(A|F)*5/9. However we still don't know P(A|F), which could be anything. –  Bogdanovist Jun 22 '12 at 4:04
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I believe now that, if we assume a ratio of men and women in the population at large, then there is a single indisputable answer.

A = the event that the person has long hair

B = the event that the person has blood type AX3

C = the event that person is female

P(C|A) = 0.9

P(C|B) = 0.8

P(C) = 0.5 (i.e. let's assume an equal ratio of men and women in the population at large)

Then P(C|A and B) = [P(C|A) x P(C|B) / P(C)] / [[P(C|A) x P(C|B) / P(C)] + [[1-P(C|A)] x [1-P(C|B)] / [1-P(C)]]]

in this case, P(C|A and B) = 0.972973

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P[C|A and B)= P(A and B and C)/P(A and B)=P(A and B and C)/ [P(A|B) P(B)]. How did you get your formula? –  Michael Chernick Jun 21 '12 at 10:12
There is probably a way to add conditions so that you get a unique answer. –  Michael Chernick Jun 21 '12 at 10:19
To add by independence of A and B the formula simplifies to P(A and B and C}/[P(A) P(B)]=P(B and C|A)/P(B). –  Michael Chernick Jun 21 '12 at 10:22
The intent of my question was really for you to justify the formula. I don't understand how it would be derived. –  Michael Chernick Jun 21 '12 at 11:24
No, the answer that supposedly used Bayes Rule is incorrect. I'm not sure why you are confused, MC's formula above is correct and cannot be used to get any result, that's what his and Whuber's answers to the question explained! –  Bogdanovist Jun 22 '12 at 4:08

Note: In order to get a definitive answer, the below answers assume that the probability of a person, a long-haired man, and a long-haired women having AX3 are approximately the same. If more accuracy is desired, this should be verified.

You start out with the knowledge that the person has long hair, so at this point the odds are:

90:10


Note: The ratio of males to females in the general population does not matter to us once we find out the person has long hair. For example, if there were 1 female in a hundred in the general population, a randomly-selected long-haired person would still be a female 90% of the time. The ratio of females to males DOES matter! (see the update below for details)

Next, we learn that the person has AX3. Because AX3 is unrelated to long hair, the ratio of men to women is known to be 50:50, and because of our assumption of the probabilities being the same, we can simply multiply each side of the probability and normalize so that the sum of the sides of the probability equals 100:

(90:10) * (80:20)
==> 7200:200

Normalize by dividing each side by (7200+200)/100 = 74

==> 7200/74:200/74
==> 97.297.. : 2.702..


Thus, the chance that the person behind the curtain is female is approximately 97.297%.

UPDATE

Here's a further exploration of the problem:

Definitions:

f - number of females
m - number of males
fl - number of females with long hair
ml - number of males with long hair
fx - number of females with AX3
mx - number of males with AX3
flx - number of females with long hair and AX3
mlx - number of males with long hair and AX3
pfl - probability that a female has long hair
pml - probability that a male has long hair
pfx - probability that a female has AX3
pmx - probability that a male has AX3


First, we are given that 90% of long-haired people are females, and 80% of people with AX3 are female, so:

fl = 9 * ml
pfl = fl / f
pml = ml / m
= fl / (9 * m)

fx = 4 * mx
pfx = fx / f
pmx = mx / m
= fx / (4 * m)


Because we assumed that the probability of AX3 is independent of gender and long hair, our calculated pfx will apply to women with long hair, and pmx will apply to men with long-hair to find the number of them that likely have AX3:

flx = fl * pfx
= fl * (fx / f)
= (fl * fx) / f
mlx = ml * pmx
= (fl / 9) * (fx / (4 * m))
= (fl * fx) / (36 * m)


Thus, the likely ratio of the number of females with long-hair and AX3 to the number of males with long-hair and AX3 is:

flx             :   mlx
(fl * fx) / f   :   (fl * fx) / (36 * m)
1/f             :   1 / (36m)
36m             :   f


Because it is given that there is an equal number of 50:50, you can cancel both sides and end with 36 females to every male. Otherwise, there are 36*m/f females for every male in the specified subgroup. For example, if there were twice as many women as men, there would be 72 females to each male of those that have long-hair and AX3.

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This solution relies on assuming more than is currently stated in the problem: namely, that long hair, AX3, and gender are independent. Otherwise, you cannot justify "applying" pfx to women with long hair, etc. –  whuber Jun 21 '12 at 19:35
@whuber: Yes, I do make that assumption. However, isn't the purpose of probability to give the best approximation based on the data that you have? Thus, since you already know that long-hair and AX3 are independent for the general population, you SHOULD carry forward that assumption to males and females until you explicitly learn otherwise. Granted, it is not a universally correct one, but it is the best one you can make until you get more info. Q: With only the current data, if you had to give the % chance that it was a woman behind the curtain, would you really say "between 0 and 100%"? –  Briguy37 Jun 21 '12 at 21:44
We have an important difference in philosophy, @Briguy. I strongly believe in not making unfounded assumptions. It is not clear in what sense the mutual independence assumption is "best": I will grant it may be in certain applications. But in general, that seems dangerous to me. I would prefer being clear about the assumptions needed to solve a problem, so people can decide whether it is worthwhile collecting the data to check those assumptions, rather than assuming things that are mathematically convenient for the sake of obtaining an answer. That's the difference between stats and math. –  whuber Jun 21 '12 at 23:14
To answer your question: yes, 0% - 100% is exactly the answer I would give. (I have given similar answers to comparable questions on this site.) That range accurately reflects the uncertainty. This issue is closely related to the Ellsberg paradox. Ellsberg's original paper is well written and clear: I recommend it. –  whuber Jun 21 '12 at 23:16
@whuber: Thanks for taking the time to dialogue with me. I see your point about the importance of thinking through and listing the assumptions made, and have updated my answer accordingly. However, in regards to your answer, I believe it is incomplete. The reason for this is that you can consider all unknown cases and find the average probability of across all of them to arrive at your final answer. E.G. Though both are still possible, probabilities above 50% are much more prevalent than probabilities below 50% across all cases, so we are surely better off guessing that it is a woman. –  Briguy37 Jun 22 '12 at 13:56
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98% Female, simple interpolation. First premise 90% female, leaves 10%, second premise only leaves 2% of the existing 10%, hence 98% female

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