LemonmathSelf-study sheet

Grade 12 · Lesson 6.1

Conditional probability

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Goals

Understand and find the conditional probability $P(A\mid B)$ , by the formula and by counting.
Read conditional probabilities from two-way tables and tree diagrams.
Use the multiplication rule $P(A\cap B)=P(B)\cdot P(A\mid B)$ .
Recognise independent events through the equation $P(A\mid B)=P(A)$ .

Remember

$P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}$ (with $P(B)>0$ ); for equally likely outcomes, $\dfrac{n(A\cap B)}{n(B)}$ .
Knowing that $B$ happened shrinks the sample space to $B$ ; in a two-way table, read only the row or column of $B$ .
$P(A\mid B)$ and $P(B\mid A)$ are usually different.
Multiplication rule: $P(A\cap B)=P(B)\cdot P(A\mid B)$ ; on a tree diagram, multiply along a path.
$A$ , $B$ independent $\iff P(A\mid B)=P(A)\iff P(A\cap B)=P(A)P(B)$ .

Exercises

Exercise 1

A fair die is rolled. Given that the result is odd, find the probability that it is at most $3$ .

Exercise 2

A class has $40$ students: $20$ boys, $14$ of whom like maths, and $20$ girls, $10$ of whom like maths. One student is chosen at random. Find the probability that the student likes maths given that she is a girl, and the probability that the student is a boy given that the student likes maths.

Exercise 3

Given $P(A)=0.5$ , $P(B)=0.4$ and $P(A\cap B)=0.2$ , find $P(A\mid B)$ and $P(B\mid A)$ , and decide whether $A$ and $B$ are independent.

Exercise 4

A box holds $4$ white balls and $6$ black balls. Two balls are drawn one after the other without replacement. Find the probability that both are black, and the probability that the second is white given that the first is black.

Exercise 5

A box holds $10$ cards numbered $1$ to $10$ . One card is drawn at random. Let $A$ be "the number is even" and $B$ be "the number is at least $7$ ". Is each claim true or false?

a) $P(B)=0.4$ .

b) $P(A\mid B)=\dfrac12$ .

c) $P(B\mid A)=\dfrac25$ .

d) $A$ and $B$ are not independent.

Exercise 6

Two fair dice are rolled. Given that at least one die shows a $6$ , find the probability that the total is at least $10$ . Give your answer to 2 decimal places.

Exercise 7

In a café, $70\%$ of customers order a coffee drink, and $40\%$ of those also order a cake. Also, $35\%$ of all customers order a cake. Find the probability that a customer orders both coffee and cake, and the probability that a customer ordered coffee given that the customer ordered cake.

Exercise 8 Review

Two marksmen fire independently at a target, one shot each, hitting with probabilities $0.8$ and $0.7$ . Given that the target is hit at least once, find the probability that both hit it. Give your answer to 2 decimal places.

Exercise 9 Review

A group has $5$ boys and $4$ girls. Three of them are chosen at random. Given that at least one girl is chosen, find the probability that exactly one girl is chosen.

Exercise 10 Challenge

A fair coin is tossed three times. Let $A$ be "the first toss is heads" and $B$ be "at least two tosses are heads". Find $P(A\mid B)$ and $P(B\mid A)$ , and decide whether $A$ and $B$ are independent.

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. Given

P(A)=0.6

P(B)=0.5

and

P(A\cap B)=0.2

, the value of

P(A\mid B)

0.3

.B.

0.4

.C.

\dfrac13

.D.

0.1

Question 2. A fair die is rolled. Given that the result is even, the probability that it is a

6

\dfrac16

.B.

\dfrac12

.C.

\dfrac13

.D.

\dfrac1{12}

Question 3. A box holds

3

red and

2

blue balls. Two balls are drawn one after the other without replacement. The probability that both are red is

\dfrac{9}{25}

.B.

\dfrac{3}{10}

.C.

\dfrac35

.D.

\dfrac12

Question 4. Of

100

people asked,

40

use app X and

30

of those are satisfied;

60

do not use X and

30

of those are satisfied. A satisfied person is chosen at random. The probability that this person uses app X is

\dfrac34

.B.

\dfrac{3}{10}

.C.

\dfrac12

.D.

\dfrac35

Part II. True or false

Question 1.

A

and

B

are independent events with

P(A)=0.4

and

P(B)=0.5

a) $P(A\cap B)=0.2$ .	T / F
b) $P(A\mid B)=0.4$ .	T / F
c) $P(\overline{A}\mid B)=0.4$ .	T / F
d) $P(A\cup B)=0.7$ .	T / F

Part III. Short answer

Question 1. Two fair dice are rolled. Given that the total is

7

, find the probability that one die shows a

1

. Give your answer to 2 decimal places.

Question 2. Of

500

people tested,

20

have a disease and

18

of them test positive;

480

do not have it and

24

of them test positive. A person who tests positive is chosen at random. Find the probability that this person has the disease. Give your answer to 2 decimal places.

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $P(A\mid B)=\dfrac{n(A\cap B)}{n(B)}$ when outcomes are equally likely.

Steps:

List $B=\{1, 3, 5\}$ .
Count the elements of $B$ that are at most $3$ .

Exercise 2.

What you need: make a two-way table; read only the row (or column) of the condition.

Steps:

The number who like maths is $14+10=24$ .
Compute $\dfrac{10}{20}$ and $\dfrac{14}{24}$ .

Exercise 3.

What you need: the definition of conditional probability; independence means $P(A\mid B)=P(A)$ .

Steps:

Divide $P(A\cap B)$ by $P(B)$ , then by $P(A)$ .
Compare $P(A\mid B)$ with $P(A)$ .

Exercise 4.

What you need: the multiplication rule $P(A\cap B)=P(A)\cdot P(B\mid A)$ ; after the first draw, $9$ balls remain.

Steps:

$P(\text{black first})=\dfrac{6}{10}$ and $P(\text{black second}\mid\text{black first})=\dfrac59$ .
If the first is black, $4$ white remain out of $9$ .

Exercise 5.

What you need: $P(A\mid B)=\dfrac{n(A\cap B)}{n(B)}$ ; independence means $P(A\mid B)=P(A)$ .

Steps:

List $B=\{7, 8, 9, 10\}$ and $A\cap B=\{8, 10\}$ .
Compare $P(A\mid B)$ with $P(A)=\dfrac12$ .

Exercise 6.

What you need: counting in the shrunken sample space.

Steps:

Count the outcomes with at least one $6$ : $36-25=11$ .
Among them, count those with total $\ge10$ .

Exercise 7.

What you need: " $40\%$ of those" is the conditional probability $P(\text{cake}\mid\text{coffee})=0.4$ .

Steps:

Use the multiplication rule.
Divide by $P(\text{cake})=0.35$ .

Exercise 8.

What you need: the multiplication rule for independent events (grade 11) and the complementary event: $P(\text{at least one})=1-P(\text{nobody hits})$ .

Steps:

$P(\text{both hit})=0.8\cdot0.7$ .
$P(\text{at least one hit})=1-0.2\cdot0.3$ ; "both hit" lies inside "at least one hit".

Exercise 9.

What you need: counting with combinations (see permutations, arrangements and combinations).

Steps:

The number of choices with at least one girl is $\binom{9}{3}-\binom{5}{3}$ .
The number with exactly one girl is $\binom{4}{1}\cdot\binom{5}{2}$ .

Exercise 10.

What you need: list the $8$ equally likely outcomes; independence means $P(A\mid B)=P(A)$ .

Steps:

Write $B$ (H for heads, T for tails): HHT, HTH, THH, HHH.
Count $A\cap B$ , then divide by $n(B)$ and by $n(A)=4$ .

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