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SL 4.11—Conditional and independent probabilities, test for independence - IB Questionbank | RevisionDojo

Practice SL 4.11—Conditional and independent probabilities, test for independence 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 game involves drawing three cards from a standard deck of 52 cards (4 suits, 13 ranks) without replacement. Let A be the event that all three cards are of the same suit, and B be the event that at least two cards are of the same rank.

Find $P(A)$ .

[3]

Find $P(B)$ .

[4]

Find $P(A \cap B)$ , and determine whether A and B are independent.

[5]

Question 2

HLPaper 2

A continuous random variable $X$ has probability density function

f(x)=\begin{cases} k\,x\,e^{-x/3},& x\ge 0,\\ 0,& x<0. \end{cases}

Determine the value of the constant $k$ .

[2]

Find $E(X)$ and $\mathrm{Var}(X)$ .

[3]

Show that the mode of $X$ is $3$ . Then, by solving $\displaystyle \int_{0}^{m} f(x)\,dx=\tfrac12$ , find the median $m$ correct to three significant figures (calculator required).

[3]

Let $Y=2X+5$ . Find $E(Y)$ and $\mathrm{Var}(Y)$ . Hence compute $P(10\le Y\le 20)$ .

[3]

Find $P(X>6\mid X>3)$ and give your answer in exact form.

[2]

Question 3

HLPaper 2

A box contains two types of biased coins. One coin is drawn at random and tossed $n=8$ times.

With probability $0.4$ , a Type A coin is drawn; for Type A, the probability of heads on any toss is $p_A=0.35$ .
With probability $0.6$ , a Type B coin is drawn; for Type B, the probability of heads on any toss is $p_B=0.65$ .

Let $X$ be the total number of heads in the 8 tosses. Assume tosses are independent conditional on the coin type.

Write down the conditional distributions of $X\mid A$ and $X\mid B$ . Hence find $P(X=3)$ .

[4]

Find $E(X)$ and $\mathrm{Var}(X)$ without expanding the full mixture PMF, by using the laws of total expectation and total variance.

[5]

The experiment resulted in exactly $x=6$ heads. Compute $P(B\mid X=6)$ .

[3]

A game pays € $g(x)=2x-9$ . Decide whether to Play (before seeing any data).

[2]

After observing that the first two tosses are both heads, you may Play the same game on the remaining six tosses. Compute $E[g(X)\mid \text{first two are heads}]$ and decide.

[4]

Question 4

SLPaper 1

A school has 20 students: 8 take Biology, 7 take Chemistry, and 5 take Physics. Some students take multiple subjects, with 3 taking Biology and Chemistry, 2 taking Biology and Physics, 1 taking Chemistry and Physics, and 1 taking all three. A committee of 4 students is formed randomly.

Find the probability that all 4 students take Biology.

[3]

Find the probability that the committee includes at least one student from each subject.

[5]

Find the probability that all 4 students take at least one of the three subjects.

[4]

Question 5

SLPaper 1

A game involves rolling two fair six-sided dice. Let A be the event that the sum of the dice is 7 , and B be the event that at least one die shows a 4 .

Find $P(A)$ .

[2]

Find $P(B)$ .

[2]

Determine whether events A and B are independent.

[3]

Question 6

SLPaper 1

Let $X$ and $Y$ be two independent events such that $P\left(X \cap Y^{\prime}\right)=0.12$ and $P\left(X^{\prime} \cap Y\right)=0.32$ .

Given that $P(X \cap Y)=k$ , find the value of $k$ .

[4]

Find $P\left(X^{\prime} \cup Y^{\prime}\right)$ . [

[2]

Question 7

SLPaper 1

A factory produces two types of components: Type A and Type B. The probability that a component is Type A is 0.6 . If it is Type A , the probability it is defective is 0.05 ; if it is Type B, the probability it is defective is 0.1 . A quality control test is conducted on components, and if a component is defective, the test detects it with probability 0.9 ; if it is not defective, the test incorrectly flags it as defective with probability 0.02 .

Draw a tree diagram to represent the probabilities of component type, defect status, and test outcome.

[4]

Find the probability that a randomly selected component tests positive for being defective.

[4]

Given that a component tests positive, find the probability that it is actually defective.

[4]

Determine whether the events "component is defective" and "component tests positive" are independent.

[3]

Question 8

HLPaper 2

A store runs a promotion in which each customer plays one spin game. Before each spin, a wheel type is chosen at random: Wheel A with probability $0.60$ and Wheel B with probability $0.40$ . The payout $Y$ (in euros) from a spin depends on the wheel type as follows (spins are independent between customers).

$\begin{array}{c|cccc} \text{Wheel A: }y & 0 & 5 & 20 & 100\\ \hline P(Y=y\mid A) & 0.70 & 0.20 & 0.09 & 0.01 \end{array}$

$\begin{array}{c|cccc} \text{Wheel B: }y & 0 & 5 & 20 & 100\\ \hline P(Y=y\mid B) & 0.50 & 0.35 & 0.10 & 0.05 \end{array}$

Find the unconditional probability distribution of $Y$ (i.e., $P(Y=y)$ for $y\in\{0,5,20,100\}$ .)

[3]

Hence compute $E(Y)$ and $\mathrm{Var}(Y)$ .

[4]

Define a “big win” as receiving at least $€20$ . For $N=40$ independent customers, find the expected number of big wins, and use technology (or a normal approximation with continuity correction) to estimate the probability that at least 8 customers get a big win.

[4]

A randomly chosen customer received $€100$ . Find the probability that their spin used Wheel B.

[3]

The store plans to charge an entry fee of $c$ euros per spin. Find the value of $c$ that makes the game fair on average for the store over many customers.

[2]

Suppose the store charges the fair price from part (e). Let a customer’s net gain be $G=Y-c$ and the store’s profit on that customer be $\Pi=c-Y=-G$ . For $N=200$ independent customers, find $E(\Pi_{\text{tot}})$ and $\mathrm{Var}(\Pi_{\text{tot}})$ , where $\Pi_{\text{tot}}=\sum_{i=1}^{200}\Pi_i$ . Use a normal approximation to determine the probability that the store loses money (that is, $\Pi_{\text{tot}}<0$ ) on that day.

[4]

Question 9

SLPaper 1

Let A and B be independent events such that $P(A)=0.6$ and $P(B)=0.4$ .

Find $P(A \cap B)$ .

[2]

Find $P\left(A^{\prime} \cap B^{\prime}\right)$ .

[2]

Find $P(A \cup B)$ .

[2]

Question 10

SLPaper 1

A bag contains 5 red balls ( R ) and 3 blue balls ( B ). Two balls are drawn without replacement.

Complete the tree diagram below by writing probabilities in the spaces provided.

[3]

Find the probability that exactly one ball is red.

[3]

SL 4.11—Conditional and independent probabilities, test for independence Questionbank