Chapter 13 · Probability

Miscellaneous · Q7: Heart attack: the patient followed meditation and yoga, $P = \frac{14}{29}$

Miscellaneous ExerciseBayes' theorem · Section 13.5

The question

Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?

Exam answer

Let $E_1$ ⁠: the patient followed the course of meditation and yoga, $E_2$ ⁠: the patient took the drug, and $A$ ⁠: the patient suffers a heart attack.

$E_1$ and $E_2$ form a partition of the sample space (each patient chooses exactly one of the two options).

$P(E_1) = P(E_2) = \frac12$

The risk of a heart attack is $40\% = \frac{40}{100}$ ⁠. The course reduces it by 30%, so

\begin{aligned} P(A|E_1) &= \frac{40}{100}\left(1 - \frac{30}{100}\right) \\ &= \frac{40}{100}\times\frac{70}{100} = \frac{28}{100} \end{aligned}

The drug reduces it by 25%, so

\begin{aligned} P(A|E_2) &= \frac{40}{100}\left(1 - \frac{25}{100}\right) \\ &= \frac{40}{100}\times\frac{75}{100} = \frac{30}{100} \end{aligned}

By the theorem of total probability,

\begin{aligned} P(A) &= P(E_1)P(A|E_1) \\ &\quad + P(E_2)P(A|E_2) \\ &= \frac12\cdot\frac{28}{100} + \frac12\cdot\frac{30}{100} \\ &= \frac{14}{100} + \frac{15}{100} = \frac{29}{100} \end{aligned}

By Bayes' theorem,

\begin{aligned} P(E_1|A) &= \frac{P(E_1)P(A|E_1)}{P(A)} \\ &= \frac{14}{100} \div \frac{29}{100} \\ &= \frac{14}{100} \times \frac{100}{29} \end{aligned}

$\therefore$ The probability that the patient followed the course of meditation and yoga is $\dfrac{14}{29}$ ⁠.

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

The one idea this problem is built on

First turn "reduces the risk by 30%" into a probability: 30% of the 40% risk is removed, leaving 28%; the drug leaves 30%. A heart attack after the yoga course is therefore a little less likely than after the drug, so seeing a heart attack makes "yoga" a little less likely than $\frac12$ ⁠. Bayes' theorem gives the exact value, $\frac{14}{29}$ ⁠.

Given, to find, and the plan

Given: The risk of a heart attack is 40%. Meditation and yoga cut the risk by 30%; the drug cuts it by 25%. Each patient chooses one option, each with probability $\frac12$ ⁠. The patient selected had a heart attack.
To find: P(the patient followed meditation and yoga | heart attack) $= P(E_1|A)$ ⁠.
Plan: (1) Name the events; check the partition. (2) Work out the reduced risks $P(A|E_1)$ and $P(A|E_2)$ ⁠. (3) Total probability for $P(A)$ ⁠. (4) Bayes' theorem. (5) Sense check.

Basics you need

1Reducing by a percentage

Reducing an amount by 30% leaves $1 - \frac{30}{100} = \frac{70}{100}$ of it. So a 40% risk reduced by 30% is $\frac{40}{100} \times \frac{70}{100} = \frac{28}{100}$ ⁠. Basics: working with fractions

2Conditional probability (Section 13.2)

The reduced risk is a probability given the option: $P(A|E_1)$ for yoga, $P(A|E_2)$ for the drug. Lesson: conditional probability

3Total probability and Bayes' theorem (Section 13.5)

$P(A) = \sum P(E_i)P(A|E_i)$ and $P(E_1|A) = \frac{P(E_1)P(A|E_1)}{P(A)}$ ⁠. Lesson: Bayes' theorem

4Tree diagrams

Branch on the option, then on heart attack or not; multiply along each path. Basics: tree diagrams

Solution

Name the events

Let
- $E_1$ ⁠: the patient followed the course of meditation and yoga;
- $E_2$ ⁠: the patient took the prescribed drug;
- $A$ ⁠: the patient suffers a heart attack.
$E_1$ and $E_2$ form a partition of the sample space: the question says each patient goes through one of the two options, and they are chosen with equal probabilities, so
$P(E_1) = P(E_2) = \frac12.$
The option chosen is the cause; the heart attack is what we observe. Note that "no option" is not a case here: every patient goes through one of the two.

Partition of a sample space (Section 13.5.1). Lesson: Bayes' theorem
The reduced risks

The risk without either option is $40\% = \frac{40}{100}$ ⁠. "Reduces the risk by 30%" means 30% of that risk is removed, so 70% of it remains:
$\begin{aligned} P(A|E_1) &= \frac{40}{100}\left(1 - \frac{30}{100}\right) \\ &= \frac{40}{100}\times\frac{70}{100} = \frac{28}{100}. \end{aligned}$
In the same way the drug leaves 75% of the risk:
$\begin{aligned} P(A|E_2) &= \frac{40}{100}\left(1 - \frac{25}{100}\right) \\ &= \frac{40}{100}\times\frac{75}{100} = \frac{30}{100}. \end{aligned}$
A reduction "by 30%" is 30% of the original amount: 30% of 40% is 12%, and $40\% - 12\% = 28\%$ ⁠. It is not 30 percentage points (that would leave $40\% - 30\% = 10\%$ ⁠); the textbook's answer $\frac{14}{29}$ confirms the first reading.

Taking a percentage of an amount. Basics: working with fractions
The chance of a heart attack: total probability

Since $E_1$ and $E_2$ form a partition, by the theorem of total probability
$\begin{aligned} P(A) &= P(E_1)P(A|E_1) \\ &\quad + P(E_2)P(A|E_2) \\ &= \frac12\cdot\frac{28}{100} + \frac12\cdot\frac{30}{100} \\ &= \frac{14}{100} + \frac{15}{100} = \frac{29}{100}. \end{aligned}$

The tree for Question 7. A heart attack comes from the yoga path (rose, $\frac{14}{100}$ ⁠) or from the drug path (green, $\frac{15}{100}$ ⁠); together $\frac{29}{100}$ ⁠.

A patient who has a heart attack followed exactly one of the two options, so the two path probabilities add.

Multiply along a path, add the paths. Basics: tree diagrams
Yoga, given a heart attack: Bayes' theorem

By Bayes' theorem,
$\begin{aligned} P(E_1|A) &= \frac{P(E_1)P(A|E_1)}{P(A)} \\ &= \frac{14}{100} \div \frac{29}{100} \\ &= \frac{14}{100} \times \frac{100}{29} = \frac{14}{29}. \end{aligned}$
The top is the yoga path; the bottom is both paths to a heart attack. 14 and 29 have no common factor (29 is prime), so $\frac{14}{29}$ is in lowest terms.

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

The drug route gets the rest: $P(E_2|A) = \frac{15}{100} \div \frac{29}{100} = \frac{15}{29}$ ⁠, and $\frac{14}{29} + \frac{15}{29} = 1$ ⁠. Since yoga lowers the risk more than the drug does, a heart attack points slightly away from yoga: $\frac{14}{29} \approx 0.483$ ⁠, just under $\frac12$ ⁠.

With equal chances for the two options, the answer is the ratio of the two risks, $28 : 30 = 14 : 15$ ⁠, so $\frac{14}{14 + 15}$ ⁠. The 40% base risk cancels, because both reduced risks are multiples of it: $28 : 30 = 70 : 75$ ⁠.

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

Result

The probability that the patient followed the course of meditation and yoga, given the heart attack, is $\frac{14}{29}$ ⁠. (Textbook answer: $\frac{14}{29}$ ⁠.)

Check it with numbers

Picture 200 patients: 100 follow yoga and 100 take the drug. Heart attacks: $0.28 \times 100 = 28$ after yoga and $0.30 \times 100 = 30$ after the drug, so $28 + 30 = 58$ in all. Of these, 28 followed yoga: $\frac{28}{58} \approx 0.4828$ ⁠, the same as $\frac{14}{29} \approx 0.4828$ ⁠.

Common mistakes

Subtracting the percentages. $40\% - 30\% = 10\%$ and $40\% - 25\% = 15\%$ treat "by 30%" as 30 percentage points; that gives $\frac{10}{25} = \frac25$ ⁠, not the book's answer.
Using 30% and 25% as $P(A|E_1)$ and $P(A|E_2)$ ⁠. Those are the reductions, not the risks that remain.
Answering $\frac12$ ⁠. That is $P(E_1)$ before we know about the heart attack.

What to take away

"Reduces by $k\%$ " multiplies by $1 - \frac{k}{100}$ ⁠.
An option that makes the outcome less likely becomes less likely once the outcome is seen.
With equally likely causes, Bayes' theorem compares $P(A|E_1)$ with $P(A|E_2)$ directly.

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)
Reducing by $k\%$ multiplies by $1 - \frac{k}{100}$	Step 2	Arithmetic (Basics: working with fractions)
Conditional probability $P(A\|E)$	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)

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Miscellaneous · Q7: Heart attack: the patient followed meditation and yoga, P=1429P = \frac{14}{29}P=2914​

Basics you need

Solution

Name the events

The reduced risks

The chance of a heart attack: total probability

Yoga, given a heart attack: Bayes' theorem

A quick sense check

Every fact used, and where it comes from

Miscellaneous · Q7: Heart attack: the patient followed meditation and yoga, $P = \frac{14}{29}$