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 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 2

SLPaper 1

In a game show, a contestant chooses one of three doors. 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 $\frac{5}{12}$ . After choosing a door, the contestant can either stay with their choice or switch to another door. If the contestant switches, the probability of winning is $\frac{7}{12}$ .

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 3

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 4

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 $P(R'\mid P)$ .

[3]

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

[2]

With $400$ applicants, find the expected number with neither.

[4]

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]

Question 5

SLPaper 2

In a factory, each item may have a cosmetic defect $A$ and/or a functional defect $B$ . From long-run monitoring: $P(A)=0.30, P(B)=0.25, P(A\cup B)=0.46.$

Determine $P(A\cap B)$ , $P(A\text{ only})$ , $P(B\text{ only})$ , and $P(\text{neither }A\text{ nor }B)$ .

[3]

Find $P(B\mid A)$ and $P(A'\mid B)$ .

[4]

Test whether $A$ and $B$ are independent, justifying your answer numerically.

[2]

A shipment of 1000 items has proportions matching these probabilities exactly. Two items are selected at random without replacement.

Find the probability that both selected items are free of defects $A$ and $B$ .

[2]

Find the probability that exactly one of the two selected items has defect $B$ (regardless of whether it also has defect $A$ ).

[2]

Three items are selected with replacement. Find the probability that at least one of the three items has both defects $A$ and $B$ .

[2]

Two items are selected without replacement. Let $C$ be the event that the sample contains at least one item with defect $A$ and no items with defect $B$ . Find $P(C)$ .

[3]

Question 6

SLPaper 1

In a group of 60 students, 35 study French, 30 study Spanish, and 10 study both languages. Let $F$ represent the event "studies French" and $S$ represent the event "studies Spanish". The following Venn diagram shows these events, with $x, y$ , and $z$ representing numbers of students. $z$

Find the values of $x, y$ , and $z$ .

[3]

Calculate the probability that a randomly selected student studies exactly one language.

[2]

Determine whether the events $F$ and $S$ are independent.

[3]

Question 7

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 8

HLPaper 2

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

Find ${\text{P}}(Y)$ .

[2]

Find $P(X)$ .

[2]

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

[2]

Question 9

SLPaper 2

Occurrences X and Y are independent and P(X) = 3P(Y). Given that P(X ∪ Y) = 0.68, find P(Y).

Find P(Y).

[6]

Question 10

HLPaper 1

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

Show that if $A$ and $B$ are mutually exclusive where $P(A\cup B)<1$ , then $A'$ and $B'$ are not mutually exclusive.

[6]