SL 4.7—Discrete random variables Questionbank

Practice SL 4.7—Discrete random variables 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

A store places a coupon in 30

Verify that $P(T=3)=0.027$ .

Find $E(T)$ .

Determine the minimum number of products to purchase so that the probability of getting at least 3 coupons is greater than 0.6 .

Question 2

SLPaper 1

A wildlife conservation program tags rare fish in a coral reef, where each fish has a 10

Verify that $P(X=4)=0.0001$ and find $P(X=5)$ .

[2]

Show that the expected number of fish checked, $E(X)$ , is 40 .

[4]

Prove that $\frac{P(X=n)}{P(X=n-1)}=\frac{0.9(n-1)(n-2)(n-3)}{6(n-4)}$ for $n>4$ .

[4]

Find the minimum number of fish that must be checked to ensure the probability of tagging at least four fish is at least 0.8 .

[4]

Question 3

SLPaper 1

A shop offers a prize if a customer collects two tokens, which are randomly placed in 20

Show that $P(Y=2)=0.04$ and find $P(Y=3)$ .

[2]

Find the expected number of products, $E(Y)$ .

[3]

Sketch the probability distribution for $n=2,3,4,5$ . A1 A1

[2]

Question 4

HLPaper 2

A biased die is weighted such that the probability, $p$ , of obtaining a six is $0.6$ . The die is rolled repeatedly and independently until a six is obtained.

Let $E$ be the event "obtaining the first six on an even numbered roll".

Find $P(E)$ .

[6]

Question 5

SLPaper 2

A discrete random variable $W$ models the number of customers arriving at a counter in a 10-minute period, with the probability distribution:

$w$	0	1	4
$P(W=w)$	0.2	$p$	$0.3 p$

(a) Find the value of $p$ and the probability distribution. (b) Find $E(W)$ and sketch the probability distribution.

Find the value of $p$ and the probability distribution.

[3]

Find $E(W)$ and sketch the probability distribution.

[3]

Question 6

HLPaper 1

Chloe and Selena play a game where each have four cards showing capital letters A, B, C and D.

Chloe lays her cards face up on the table in order A, B, C, D as shown in the following diagram.

Selena shuffles her cards and lays them face down on the table. She then turns them over one by one to see if her card matches with Chloe’s card directly above.

Chloe wins if no matches occur; otherwise Selena wins.

Chloe and Selena repeat their game so that they play a total of 50 times.

Suppose the discrete random variable $X$ represents the number of times Chloe wins.

Show that the probability that Chloe wins the game is $\frac{3}{8}$ .

[6]

Determine the mean of $X$ .

[3]

Determine the variance of $X$ .

[2]

Question 7

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 are offered the option to play the game (where the payoff $g(X)$ depends on the total number of heads $X$ in the 8 tosses). Compute $E[g(X)\mid \text{first two are heads}]$ and decide whether to play.

[4]

Question 8

SLPaper 1

A game involves rolling a fair six-sided die until a 6 appears. Let $X$ be the discrete random variable representing the number of rolls needed to get the first 6. The probability distribution is given by $P(X=n)=\frac{5^{n-1}}{6^{n}}$ for $n=1,2,3, \ldots$ .

Verify that $P(X=1)=\frac{1}{6}$ and $P(X=2)=\frac{5}{36}$ .

[2]

Show that the expected number of rolls, $E(X)$ , is 6.

[3]

Find the probability that at least 4 rolls are needed to get the first 6.

[3]

Question 9

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 $\text{Var}(Y)$ .

[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 5. 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 $\text{Var}(\Pi_{\text{tot}})$ , 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 10

SLPaper 1

A discrete random variable $X$ represents the number of successful attempts in a game, with the probability distribution given by the following table.

$x$	0	1	2	3
$P(X=x)$	$\frac{1}{5}$	$p$	$\frac{1}{10}$	$q$

Given that the expected value $E(X)=\frac{7}{5}$ ,

Find the values of $p$ and $q$ .

[5]

Calculate the variance $\operatorname{Var}(X)$ .

[3]

w

P(W=w)

0.2

p

0.3 p

x

P(X=x)

\frac{1}{5}

p

\frac{1}{10}

q

w

P(W=w)

0.2

p

0.3 p

x

P(X=x)

\frac{1}{5}

p

\frac{1}{10}

q

w

P(W=w)

0.2

p

0.3 p

x

P(X=x)

\frac{1}{5}

p

\frac{1}{10}

q