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SL 4.8—Binomial distribution - IB Questionbank | RevisionDojo

Practice SL 4.8—Binomial distribution 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 mobile app has a feature that successfully loads with probability 0.85 each time it is accessed. A user accesses the feature 8 times in a session. Let $Y$ be the number of times the feature successfully loads.

Find the probability that the feature successfully loads exactly 6 times.

[3]

Find the probability that the feature successfully loads at least 7 times.

[3]

Question 2

SLPaper 2

A factory produces light bulbs, and the probability that a bulb is defective is 0.03. A quality control inspector selects 10 bulbs at random for testing. (a) Write down the probability that a randomly selected bulb is not defective. (b) Let $X$ be the number of defective bulbs in the sample. Find the expected number of defective bulbs. (c) Find the probability that exactly one bulb in the sample is defective.

Write down the probability that a randomly selected bulb is not defective.

[1]

Let $X$ be the number of defective bulbs in the sample. Find the expected number of defective bulbs.

[2]

Find the probability that exactly one bulb in the sample is defective.

[3]

Question 3

HLPaper 2

A discrete random variable $X$ represents the number of successful attempts in a sequence of 5 independent trials, each with a success probability of 0.3. The probability distribution of $X$ follows a binomial distribution.

Write down the probability mass function of $X$ .

[2]

Calculate the expected value $E(X)$ .

[2]

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

[2]

Sketch the probability distribution of $X$ .

[3]

Question 4

SL & HLPaper 2

Two fair 4-sided dice are rolled. Each die has faces numbered from 1 to 4.

List the sample space of all the outcomes of the two dice along with their sums.

[2]

Find the probability that the sum of the numbers on the two dice is greater than or equal to 6.

[2]

Find the probability that the sum of the numbers on the two dice is smaller than 6 through a different approach than the previous part.

[2]

If these dice rolls were repeated 10 times, what would be the probability to see the sum greater than or equal to 6 exactly 5 times.

[2]

Question 5

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 6

SLPaper 2

A machine produces screws, and the probability that a screw is defective is 0.04. A quality control inspector selects 15 screws at random for testing. Let $X$ represent the number of defective screws in the sample.

Find the expected number of defective screws in the sample.

[2]

Find the probability that there are exactly 2 defective screws.

[3]

Find the probability that there are at most 2 defective screws.

[3]

If the inspector finds at least one defective screw, find the probability that there are exactly 2 defective screws.

[4]

Question 7

SLPaper 2

In a large college the probability that a learner is left handed is 0.08. A sample of 150 learners is randomly selected from the college. Let $k$ be the expected number of left-handed learners in this sample.

Find $k$ .

[2]

Hence, find the probability that exactly $k$ learners are left handed.

[2]

Hence, find the probability that fewer than $k$ learners are left handed.

[2]

Question 8

SLPaper 2

A marine biologist is studying the nesting success of sea turtles on a protected beach. Each turtle nest has a 0.65 probability of producing at least one viable hatchling, based on environmental conditions. During a nesting season, 30 nests are monitored, and the number of successful nests is denoted by $X$ . If at least 20 nests are successful, the beach qualifies for additional conservation funding. If a nest is successful, it is further evaluated for a special research grant, where each successful nest has a 0.3 probability of being selected.

Find the probability that exactly 18 nests are successful.

[3]

Calculate the expected number and variance of successful nests.

[4]

Find the probability that the beach qualifies for additional conservation funding.

[3]

Given that the beach qualifies for funding, find the probability that exactly 22 nests are successful.

[4]

In the special research grant evaluation, let $Y$ represent the number of successful nests selected for the grant. Find the expected number of nests selected, given that the beach qualifies for funding.

[5]

The biologist wants to ensure that the probability of qualifying for funding is at least 0.9 by increasing the number of monitored nests, $n$ . Find the smallest value of $n$ .

[5]

Question 9

HLPaper 2

The random variable $Y$ has a binomial distribution with parameters $m$ and $q$ . It is given that $E(Y) = 3.5$ .

Find the least possible value of $m$ .

[2]

It is further given that $P(Y \leq 1) = 0.09478$ correct to 4 significant figures.

Determine the value of $m$ and the value of $q$ .

[5]

Question 10

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]

Paper

Level

Question 1

SLPaper 2

Find the probability that the feature successfully loads exactly 6 times.

[3]

Find the probability that the feature successfully loads at least 7 times.

[3]

Question 2

SLPaper 2

Write down the probability that a randomly selected bulb is not defective.

[1]

Let $X$ be the number of defective bulbs in the sample. Find the expected number of defective bulbs.

[2]

Find the probability that exactly one bulb in the sample is defective.

[3]

Question 3

HLPaper 2

Write down the probability mass function of $X$ .

[2]

Calculate the expected value $E(X)$ .

[2]

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

[2]

Sketch the probability distribution of $X$ .

[3]

Question 4

SL & HLPaper 2

Two fair 4-sided dice are rolled. Each die has faces numbered from 1 to 4.

List the sample space of all the outcomes of the two dice along with their sums.

[2]

Find the probability that the sum of the numbers on the two dice is greater than or equal to 6.

[2]

Find the probability that the sum of the numbers on the two dice is smaller than 6 through a different approach than the previous part.

[2]

If these dice rolls were repeated 10 times, what would be the probability to see the sum greater than or equal to 6 exactly 5 times.

[2]

Question 5

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 6

SLPaper 2

Find the expected number of defective screws in the sample.

[2]

Find the probability that there are exactly 2 defective screws.

[3]

Find the probability that there are at most 2 defective screws.

[3]

If the inspector finds at least one defective screw, find the probability that there are exactly 2 defective screws.

[4]

Question 7

SLPaper 2

Find $k$ .

[2]

Hence, find the probability that exactly $k$ learners are left handed.

[2]

Hence, find the probability that fewer than $k$ learners are left handed.

[2]

Question 8

SLPaper 2

Find the probability that exactly 18 nests are successful.

[3]

Calculate the expected number and variance of successful nests.

[4]

Find the probability that the beach qualifies for additional conservation funding.

[3]

Given that the beach qualifies for funding, find the probability that exactly 22 nests are successful.

[4]

[5]

The biologist wants to ensure that the probability of qualifying for funding is at least 0.9 by increasing the number of monitored nests, $n$ . Find the smallest value of $n$ .

[5]

Question 9

HLPaper 2

The random variable $Y$ has a binomial distribution with parameters $m$ and $q$ . It is given that $E(Y) = 3.5$ .

Find the least possible value of $m$ .

[2]

It is further given that $P(Y \leq 1) = 0.09478$ correct to 4 significant figures.

Determine the value of $m$ and the value of $q$ .

[5]

Question 10

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]

[4]

Paper

Level

Question 1

SLPaper 2

Find the probability that the feature successfully loads exactly 6 times.

[3]

Find the probability that the feature successfully loads at least 7 times.

[3]

Question 2

SLPaper 2

Write down the probability that a randomly selected bulb is not defective.

[1]

Let $X$ be the number of defective bulbs in the sample. Find the expected number of defective bulbs.

[2]

Find the probability that exactly one bulb in the sample is defective.

[3]

Question 3

HLPaper 2

Write down the probability mass function of $X$ .

[2]

Calculate the expected value $E(X)$ .

[2]

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

[2]

Sketch the probability distribution of $X$ .

[3]

Question 4

SL & HLPaper 2

Two fair 4-sided dice are rolled. Each die has faces numbered from 1 to 4.

List the sample space of all the outcomes of the two dice along with their sums.

[2]

Find the probability that the sum of the numbers on the two dice is greater than or equal to 6.

[2]

Find the probability that the sum of the numbers on the two dice is smaller than 6 through a different approach than the previous part.

[2]

If these dice rolls were repeated 10 times, what would be the probability to see the sum greater than or equal to 6 exactly 5 times.

[2]

Question 5

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 6

SLPaper 2

Find the expected number of defective screws in the sample.

[2]

Find the probability that there are exactly 2 defective screws.

[3]

Find the probability that there are at most 2 defective screws.

[3]

If the inspector finds at least one defective screw, find the probability that there are exactly 2 defective screws.

[4]

Question 7

SLPaper 2

Find $k$ .

[2]

Hence, find the probability that exactly $k$ learners are left handed.

[2]

Hence, find the probability that fewer than $k$ learners are left handed.

[2]

Question 8

SLPaper 2

Find the probability that exactly 18 nests are successful.

[3]

Calculate the expected number and variance of successful nests.

[4]

Find the probability that the beach qualifies for additional conservation funding.

[3]

Given that the beach qualifies for funding, find the probability that exactly 22 nests are successful.

[4]

[5]

The biologist wants to ensure that the probability of qualifying for funding is at least 0.9 by increasing the number of monitored nests, $n$ . Find the smallest value of $n$ .

[5]

Question 9

HLPaper 2

The random variable $Y$ has a binomial distribution with parameters $m$ and $q$ . It is given that $E(Y) = 3.5$ .

Find the least possible value of $m$ .

[2]

It is further given that $P(Y \leq 1) = 0.09478$ correct to 4 significant figures.

Determine the value of $m$ and the value of $q$ .

[5]

Question 10

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]

[4]

SL 4.8—Binomial distribution Questionbank