SL 4.6—Combined, mutually exclusive, conditional, independence, prob diagrams Questionbank

Practice SL 4.6—Combined, mutually exclusive, conditional, independence, prob diagrams with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 1

In a school club of 40 students, 25 students participate in debates, 15 students participate in quizzes, and 8 students participate in both activities. The following Venn diagram represents the events $D$ (participates in debates) and $Q$ (participates in quizzes), with $d$ and $q$ representing numbers of students.

Find the value of $d$ .

[2]

Find the value of $q$ .

[2]

Calculate the probability that a randomly selected student participates in exactly one of these activities.

[3]

Question 2

SLPaper 1

In a factory, 8

Write down the probability that a product is not defective.

Find the probability that both products are not defective.

Find the probability that exactly one of the two products is defective.

Determine whether the events "first product is defective" and "second product is defective" are independent.

Question 3

SLPaper 1

Let $A$ and $B$ be two events

Let $P(A) = P(B) = 0.4$ and $P(A \cap B) = 0.2$ . Find $P(B|A)$ .

[2]

Find $P(A|B)$ .

[1]

Show that for any two events $A$ and $B$ where $P(A) = P(B)$ , the conditional probabilities $P(A|B)$ and $P(B|A)$ are equal.

[1]

Question 4

SLPaper 2

A store sells two types of coffee: Arabica and Robusta. The probability that a customer buys Arabica is $\frac{3}{5}$ . If a customer buys Arabica, the probability they purchase a large size is $\frac{2}{3}$ . If they buy Robusta, the probability they purchase a large size is $\frac{1}{4}$ . The following tree diagram represents these events.

Complete the tree diagram by finding the values of $p, q, r$ , and $s$ .

[2]

Find the probability that a customer buys a large coffee.

[3]

Given that a customer buys a large coffee, find the probability they purchased Arabica.

[3]

Question 5

HLPaper 2

Events $X$ and $Y$ are such that $P(X \cup Y) = 0.95$ , $P(X \cap Y) = 0.6$ and $P(X|Y) = 0.75$ .

Find $P(Y)$ .

[2]

Find $P(X)$ .

[2]

Hence show that events $X'$ and $Y$ are independent.

[2]

Question 6

SLPaper 1

In a game show, a contestant chooses one of three doors at random. The probability that a prize is behind door A is $\frac{1}{4}$ , behind door B is $\frac{1}{3}$ , and behind door C is $\frac{5}{12}$ . The host, who knows where the prize is, opens one of the other doors which does not contain the prize. The contestant can then either stay with their choice or switch to the remaining closed door.

Calculate the probability that the contestant wins if they stay with their initial choice.

[3]

Determine whether the events "choosing door A" and "winning by switching" are independent.

[3]

Question 7

SLPaper 2

Two urns contain red (R) and blue (B) balls:

Urn A: 5 R, 7 B (12 balls)
Urn B: 4 R, 6 B (10 balls)

An experiment proceeds as follows (independently each time it is run).

With probability $0.6$ (Procedure I): draw two balls without replacement from Urn A.
With probability $0.4$ (Procedure II): transfer one ball chosen uniformly at random from Urn A to Urn B, then draw two balls without replacement from Urn B.

Let $E$ be the event “the two drawn balls are both red”.

Under Procedure I, find $P(\text{both red})$ and $P(\text{at least one red})$ .

[3]

Under Procedure II, find $P(\text{both red})$ .

[4]

Find the overall probability $P(E)$ .

[2]

Given that the two drawn balls are both red, find the posterior probability that Procedure II was used.

[3]

Under Procedure II, given that the two drawn balls are both blue, find the probability that the transferred ball was blue.

[3]

The experiment is performed twice independently. Find the probability that exactly one run results in both red.

[2]

Question 8

SLPaper 2

Events $X$ and $Y$ are independent and $P(X) = 3P(Y)$ . Given that $P(X \cup Y) = 0.68$ , find $P(Y)$ .

Find $P(Y)$ .

[6]

Question 9

HLPaper 1

Consider two events $A$ and $B$ that are mutually exclusive.

Given that $P(A \cup B) < 1$ , show that $A'$ and $B'$ are not mutually exclusive.

[6]

Question 10

SLPaper 2

At a tech firm’s internship intake, each applicant may have a portfolio ( $P$ ) and/or a recommendation ( $R$ ). From historical records: $P(P)=0.55,\quad P(R)=0.65,\quad P(P\cap R)=0.30$

Interview pass probabilities: $P(I\mid P\cap R)=0.80,\; P(I\mid P\cap R')=0.60,\; P(I\mid P'\cap R)=0.50,\; P(I\mid P'\cap R')=0.30$

Compute $P(P\cup R)$ and complete the four-region table.

[3]

Find $P(P'\mid R)$ and $P(R'\mid P)$ .

[4]

Test whether $P$ and $R$ are independent.

[2]

With $400$ applicants, find the expected number with neither a portfolio nor a recommendation.

[2]

Find the probability that two out of $400$ applicants chosen without replacement have a recommendation letter.

[2]

Find $P(I)$ .

[2]

Given an applicant passed and had a portfolio, find $P(R\mid P\cap I)$ .

[3]