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.
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 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.
Let be the number of defective bulbs in the sample. Find the expected number of defective bulbs.
Find the probability that exactly one bulb in the sample is defective.
A discrete random variable 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 follows a binomial distribution.
Write down the probability mass function of .
Calculate the expected value .
Find the variance .
Sketch the probability distribution of .
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 represent the number of defective screws in the sample.
Find the expected number of defective screws in the sample.
Find the probability that there are exactly 2 defective screws.
Find the probability that there are at most 2 defective screws.
If the inspector finds at least one defective screw, find the probability that there are exactly 2 defective screws.
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 . 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.
Calculate the expected number and variance of successful nests.
Find the probability that the beach qualifies for additional conservation funding.
Given that the beach qualifies for funding, find the probability that exactly 22 nests are successful.
In the special research grant evaluation, let represent the number of successful nests selected for the grant. Find the expected number of nests selected, given that the beach qualifies for funding.
The biologist wants to ensure that the probability of qualifying for funding is at least 0.9 by increasing the number of monitored nests, . Find the smallest value of .
A quality control system in a factory tests electronic components. The probability that a component passes the test is 0.88 . A batch consists of 20 components, and the number of components that pass is denoted by . If fewer than 15 components pass, the batch is rejected.
Find the probability that a batch is rejected.
Find the expected number of batches that are rejected out of 50 batches tested.
Find the probability that at most 2 out of 50 batches are rejected. Include a probability distribution graph to illustrate your answer.
Given that a batch is not rejected, find the probability that exactly 16 components pass the test.
The factory wants to adjust the number of components in a batch, , so that the variance of the number of components that pass is at least 3.5. Find the smallest value of .
A coffee shop offers a loyalty program where customers have a 0.2 probability of receiving a free coffee each visit. A customer visits the shop 12 times. Let be the number of free coffees received.
Find the expected number of free coffees.
Find the probability that the customer receives at least one free coffee.
Find the smallest number of visits, , such that the expected number of free coffees is at least 3 .
A tech company develops a facial recognition software used in security systems. Each authentication attempt has a 0.92 probability of being successful. During a high-security event, a system processes 40 authentication attempts, and the number of successful authentications is denoted by . If fewer than 35 authentications are successful, the system triggers a manual verification protocol. Additionally, successful authentications are audited, with each having a 0.15 probability of being flagged for further review.
Find the probability that exactly 36 authentications are successful.
Calculate the expected number and variance of successful authentications.
Find the probability that the manual verification protocol is triggered.
Given that the manual verification protocol is not triggered, find the probability that exactly 37 authentications are successful.
In the audit, let represent the number of successful authentications flagged for review. Find the expected number of flagged authentications, given that the manual verification protocol is not triggered.
The company wants to adjust the number of authentication attempts, , so that the variance of successful authentications is at least 4.2. Find the smallest value of .
Find the smallest number of authentication attempts, , such that the probability of triggering the manual verification protocol is less than 0.05 .
A charity raffle sells tickets, and each ticket has a 0.05 probability of winning a prize. A group of 25 tickets is purchased, and the number of winning tickets is denoted by . If at least 3 tickets win, the group qualifies for a bonus draw, where each winning ticket in the bonus draw has a 0.1 probability of winning an additional prize.
Find the probability that the group qualifies for the bonus draw.
Find the expected number and variance of winning tickets in the original raffle.
Given that the group qualifies for the bonus draw, find the probability that exactly 4 tickets win in the original raffle. Include a probability distribution graph to illustrate the conditional probability.
In the bonus draw, let represent the number of additional prizes won. Find the expected number of additional prizes won, given that the group qualifies for the bonus draw.
The charity wants to ensure that the probability of qualifying for the bonus draw is at least 0.8 by adjusting the number of tickets purchased, . Find the smallest value of .
A game show uses a spinner divided into 8 equal sections, with 3 sections colored blue and 5 sections colored white. The spinner is spun 12 times in a single game, and each spin is independent. A player earns a point for each blue section spun. Let represent the number of points earned.
Find the probability that the player earns exactly 4 points in the game.
Find the expected number of points and the variance of .
Find the probability that the player earns at least 5 points. Include a probability distribution graph to illustrate your answer.
Given that the player earns at least 5 points, find the probability that they earn exactly 6 points.
The game show organizers want to adjust the number of spins, , so that the probability of earning at least 5 points is at least 0.95 . Find the smallest value of .
A basketball player has a 0.7 probability of making a free throw. During a practice session, the player attempts 8 free throws. Let represent the number of successful free throws.
Find the variance of .
Find the probability that the player makes at least 6 free throws. Include a probability distribution graph to illustrate your answer.
Given that the player makes at least 6 free throws, find the probability that exactly 7 free throws are made.