Chapter 13 · Probability

Miscellaneous · Q3: Grey hair: the person selected is male, $P = \frac{20}{21}$

Miscellaneous ExerciseBayes' theorem · Section 13.5

The question

Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.

Exam answer

Let $E_1$ ⁠: the person selected is male, $E_2$ ⁠: the person selected is female, and $G$ ⁠: the person has grey hair.

$E_1$ and $E_2$ form a partition of the sample space (each person is either male or female).

As there are equal numbers of males and females, $P(E_1) = P(E_2) = \frac12$ ⁠.

$P(G|E_1) = 5\% = \frac{1}{20}$

$P(G|E_2) = 0.25\% = \frac{1}{400}$

By the theorem of total probability,

\begin{aligned} P(G) &= P(E_1)P(G|E_1) \\ &\quad + P(E_2)P(G|E_2) \\ &= \frac12\cdot\frac{1}{20} + \frac12\cdot\frac{1}{400} \\ &= \frac{20}{800} + \frac{1}{800} = \frac{21}{800} \end{aligned}

By Bayes' theorem,

\begin{aligned} P(E_1|G) &= \frac{P(E_1)P(G|E_1)}{P(G)} \\ &= \frac{20}{800} \div \frac{21}{800} \\ &= \frac{20}{800} \times \frac{800}{21} \end{aligned}

$\therefore$ The probability that the grey haired person is male is $\dfrac{20}{21}$ ⁠.

Want the reason behind every step, the basics and the checks?

The one idea this problem is built on

Grey hair is twenty times as common among men (5%) as among women (0.25%), and there are as many men as women. So among grey haired people, men outnumber women by 20 to 1, and a grey haired person picked at random is male with probability $\frac{20}{21}$ ⁠. Bayes' theorem turns this reasoning into a calculation.

Given, to find, and the plan

Given: 5% of men and 0.25% of women have grey hair; there are equal numbers of males and females; the person selected has grey hair.
To find: The probability that the person is male, given grey hair: $P(E_1|G)$ ⁠.
Plan: (1) Name the events; check the partition. (2) Turn the percentages into fractions. (3) Total probability for $P(G)$ ⁠. (4) Bayes' theorem. (5) Sense check.

Basics you need

1Percentages as fractions

$k\%$ means $\frac{k}{100}$ ⁠: $5\% = \frac{5}{100} = \frac{1}{20}$ and $0.25\% = \frac{0.25}{100} = \frac{1}{400}$ ⁠. Basics: working with fractions

2Conditional probability (Section 13.2)

"5% of men have grey hair" is a statement about men only: $P(G|E_1) = \frac{1}{20}$ ⁠. Lesson: conditional probability

3Total probability and Bayes' theorem (Section 13.5)

For a partition $E_1, E_2$ ⁠: $P(G) = \sum P(E_i)P(G|E_i)$ and $P(E_1|G) = \frac{P(E_1)P(G|E_1)}{P(G)}$ ⁠. Lesson: Bayes' theorem

4Tree diagrams

Multiply along each path; add the paths that end in grey hair. Basics: tree diagrams

Solution

Name the events

Let
- $E_1$ ⁠: the person selected is male;
- $E_2$ ⁠: the person selected is female;
- $G$ ⁠: the person selected has grey hair.
$E_1$ and $E_2$ form a partition of the sample space: every person is either male or female, not both.

Being male or female is the "cause"; grey hair is what we observe. The question asks for the cause given what we observe, which is what Bayes' theorem finds.

Partition: pairwise disjoint, exhaustive, nonzero probabilities (Section 13.5.1). Lesson: Bayes' theorem
The four probabilities, as fractions

Equal numbers of males and females give
$P(E_1) = P(E_2) = \frac12.$
The percentages are conditional probabilities:
$P(G|E_1) = 5\% = \frac{5}{100} = \frac{1}{20},$ $P(G|E_2) = 0.25\% = \frac{25}{10000} = \frac{1}{400}.$
"5% of men" is 5 out of every 100 men, so it is a probability given $E_1$ ⁠, not $P(G)$ ⁠. For 0.25%, multiply the top and bottom of $\frac{0.25}{100}$ by 100 to clear the decimal: $\frac{25}{10000}$ ⁠, and then divide both by 25.

$k\% = \frac{k}{100}$ ⁠. Basics: working with fractions
The chance of grey hair: total probability

Since $E_1$ and $E_2$ form a partition, by the theorem of total probability
$\begin{aligned} P(G) &= P(E_1)P(G|E_1) \\ &\quad + P(E_2)P(G|E_2) \\ &= \frac12\cdot\frac{1}{20} + \frac12\cdot\frac{1}{400} \\ &= \frac{1}{40} + \frac{1}{800} \\ &= \frac{20}{800} + \frac{1}{800} = \frac{21}{800}. \end{aligned}$

The tree for Question 3. A grey haired person comes from the man path (rose, $\frac{1}{40}$ ⁠) or from the woman path (green, $\frac{1}{800}$ ⁠). The man path is 20 times as likely.

A grey haired person is either a grey haired man or a grey haired woman, never both, so the two path probabilities add. The common denominator of 40 and 800 is 800.

Multiply along a path, add the paths. Basics: tree diagrams
Male, given grey hair: Bayes' theorem

By Bayes' theorem,
$\begin{aligned} P(E_1|G) &= \frac{P(E_1)P(G|E_1)}{P(G)} \\ &= \frac{20}{800} \div \frac{21}{800} \\ &= \frac{20}{800} \times \frac{800}{21} = \frac{20}{21}. \end{aligned}$
The top is the man path, $\frac{1}{40} = \frac{20}{800}$ ⁠; the bottom is every path to grey hair, $\frac{21}{800}$ ⁠. Out of every 21 grey haired people, on average 20 are men.

Dividing by a fraction is multiplying by its flip. Basics: dividing by a fraction
A quick sense check

The woman route gets the rest: $P(E_2|G) = \frac{1}{800} \div \frac{21}{800} = \frac{1}{21}$ ⁠, and $\frac{20}{21} + \frac{1}{21} = 1$ ⁠. Before we saw the grey hair, the chance of a man was $\frac12$ ⁠; grey hair raises it to $\frac{20}{21} \approx 0.952$ ⁠.

With equal numbers of men and women, the $\frac12$ factors cancel, so the answer depends only on the ratio $\frac{1}{20} : \frac{1}{400} = 20 : 1$ ⁠. That gives $\frac{20}{20 + 1}$ ⁠.

$P(E_1|G) + P(E_2|G) = 1$ for a partition. Lesson: Bayes' theorem

Result

The probability that the grey haired person is male is $\frac{20}{21}$ ⁠. (Textbook answer: $\frac{20}{21}$ ⁠.)

Check it with numbers

Picture 8000 people: 4000 men and 4000 women. Grey haired men: $0.05 \times 4000 = 200$ ⁠. Grey haired women: $0.0025 \times 4000 = 10$ ⁠. So $200 + 10 = 210$ people have grey hair, and 200 of them are men: $\frac{200}{210} \approx 0.9524$ ⁠, the same as $\frac{20}{21} \approx 0.9524$ ⁠.

Common mistakes

Answering 5%. That is $P(G|E_1)$ ⁠, the chance that a man has grey hair. The question asks the reverse: the chance that a grey haired person is a man.
Misreading 0.25%. It is $\frac{0.25}{100} = \frac{1}{400}$ ⁠, not $0.25 = \frac14$ and not $\frac{25}{100}$ ⁠.
Leaving out $P(E_1)$ and $P(E_2)$ ⁠. Here they are equal and cancel, but in general they change the answer; write them in.

What to take away

" $k\%$ of a group have a property" is a conditional probability, given that group.
When the causes are equally likely, Bayes' theorem reduces to comparing $P(G|E_1)$ with $P(G|E_2)$ ⁠: here 20 to 1, so $\frac{20}{21}$ ⁠.
Write every percentage as an exact fraction first; decimals inside fractions invite slips.

Every fact used, and where it comes from

Fact	Used in	Source
Partition of a sample space	Step 1	Section 13.5.1 (Lesson: Bayes' theorem)
$k\% = \frac{k}{100}$	Step 2	Arithmetic (Basics: working with fractions)
Conditional probability $P(G\|E)$	Step 2	Section 13.2 (Lesson: conditional probability)
Theorem of total probability	Step 3	Section 13.5.2 (Lesson: Bayes' theorem)
Bayes' theorem	Steps 4, 5	Section 13.5 (Examples 15 to 21)
Dividing by a fraction	Steps 4, 5	Arithmetic (Basics: dividing by a fraction)

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Miscellaneous · Q3: Grey hair: the person selected is male, P=2021P = \frac{20}{21}P=2120​

Basics you need

Solution

Name the events

The four probabilities, as fractions

The chance of grey hair: total probability

Male, given grey hair: Bayes' theorem

A quick sense check

Every fact used, and where it comes from

Miscellaneous · Q3: Grey hair: the person selected is male, $P = \frac{20}{21}$