Binomial Distribution (DP IB Analysis & Approaches (AA): HL): Exam Questions

State the conditions that must be satisfied to be able to model a random variable $X$ with a binomial distribution $\mathrm{B}(n,p)$.

A fair spinner has 8 sectors labelled with the numbers 1 through 8. For each of the following cases, state with a reason whether or not a binomial distribution would be appropriate for modelling the specified random variable.

(i) The random variable $S$ is the number of the sector that the spinner lands on when it is spun.

(ii) The random variable $W$ is the number of times the spinner is spun until it lands on ‘7’ for the first time.

(iii) The random variable $Y$ is the number of times the spinner lands on a prime number when it is spun twelve times.

(iv) On the first spin, it is a ‘win’ if the spinner lands on an even number. On subsequent spins it is a ‘win’ if the spinner lands either on the same number as the previous spin or on a factor of the number from the previous spin. The random variable $L$ is the number of wins when the spinner is spun ten times.

On any given day during a normal five-day working week, there is a 60% chance that Yussuf catches a taxi to work.

Find $\mathrm{E}\left(X\right)$, the expected number of times Yussuf will catch a taxi to work during a normal five-day working week.

Find the probability that, during a normal five-day working week, Yussuf never catches a taxi.

Find the probability that, during a normal five-day working week, Yussuf catches a taxi once at the most.

A particular medical procedure is found to have a success rate of 78%. Twelve patients are on the waiting list to undergo the procedure.

Find the probability that all twelve patients’ procedures are successful.

Find the probability that all but two of the patients have successful procedures.

For a jellyfish population in a certain area of the ocean, there is a 95% chance that any given jellyfish contains microplastic particles in its body.

For a sample size of 40 jellyfish from this population, find the probability of:

(i) exactly 38 jellyfish

(ii) all the jellyfish

having microplastic particles in their bodies.

Giovanni is rolling a biased die, for which the probability of landing on a two is 0.25. He rolls the die 10 times and records the number of times that it lands on a two. Find the probability that

(i) the die lands on a two 4 times

(ii) the die lands on a two 4 times by landing on a two 3 times in the first 9 rolls, and then landing on a two on the tenth roll.

For cans of a particular brand of soft drink labelled as containing 330 ml, the actual volume of soft drink in a can varies. Although the company’s quality control assures that the mean volume of soft drink in the cans remains at 330 ml, it is known from experience that the probability of any particular can of the soft drink containing less than 320 ml is 0.0296.

Tilly buys a pack of 24 cans of this soft drink. It may be assumed that those 24 cans represent a random sample. Let $L$ represent the number of cans in the pack that contain less than 320 ml of soft drink.

Find the probability that

(i) none of the cans

(ii) exactly two of the cans

(iii) at least two of the cans

contain less than 320 ml of soft drink.

The random variable $X~B(40,0.15)$. 

Find:

(i) $P(3\le X<14)$

(ii) $P(5<X<12)$

Find $\mathrm{Var}\left(X\right)$.

Find $P(X\le 3|X\le 9)$.

In the town of Wooster, Ohio, it is known that 90% of the residents prefer the locally produced Woostershire brand sauce when preparing a Caesar salad.  The other 10% of residents prefer another well-known brand.

30 residents are chosen at random by a pollster.  Let the random variable $X$ represent the number of those 30 residents that prefer Woostershire brand sauce.

Find

(i) $E\left(X\right)$

(ii) $\mathrm{Var}\left(\mathrm{X}\right)$

Find the probability that

(i) 90% or more of the residents chosen prefer Woostershire brand sauce,

(ii) none of the residents chosen prefer the other well-known brand.

Two fair dice are rolled and the numbers showing on the dice are added together. This is done 15 times and the number of times the sum is equal to 2, 7 or 11 is recorded. Let $X$ be the discrete random variable representing the number of times that the sum is equal to 2, 7 or 11.

Find the expected value of $X$.

Find the probability that $X$ is at least 2 but no more than 6.

On any given day, the probability that Mark rides his bike to school is $\frac{1}{3}$.

Find the probability that, in a week with five school days, Mark rides his bike to school on exactly two days. Give your answer as a fraction.

Find the probability that, in a week with five school days, Mark rides his bike to school on at least one day.

At a builders’ convention there is a competition in which 17 contestants get to choose one out of ten gift boxes. Two of the gift boxes have a chainsaw and the others have a regular saw. After a builder selects a box and claims the gift, the saw is replenished so that the next builder has the same options.

Find the probability that a contestant gets a regular saw.

Find the probability that exactly five of the 17 contestants win a chainsaw.

Find the probability that at least three and no more than nine contestants win a chainsaw.

In a game show, each contestant has seven boxes they can open. Two boxes are empty, two boxes have $10 inside, two boxes have $100 inside and one box has $10 000 inside. A contestant randomly selects a box and wins the amount that is inside.

Find the expected payoff from playing the game.

28 contestants each play the game once.

Find the expected number of players that win

(i) $10

(ii) $100

(iii) $10 000.

Find the probability that

(i) exactly four contestants win $10 000

(ii) less than six contestants win $10 000.

In a normal pack of 52 playing cards (without the jokers) there are 13 of each suit: hearts, clubs, diamonds and spades. James plays a card game with some friends. Each player is given nine full packs of cards. They randomly pick one card from each full pack so that they each have a hand of nine cards.

Find the probability that James has at least three clubs.

Hearts and diamonds are known as the red suits. Clubs and spades are known as the black suits.

Find the probability that James only has red cards in his hand.

Find the probability that James has twice as many black cards as red cards in his hand.

A biased die is rolled 24 times. The probability that the die lands on a prime number is $\frac{3}{4}$. The die is equally likely to land on any of the prime numbers. Similarly, the other numbers all have the same chance as each other of being the number that the die lands on.

Find the probability that the die lands on a prime number no more than 15 times.

Find the expected number of times the die lands on a three.

Find the expected number of times the die lands on a one.

The germination rate of a particular seed is 44%. George sows 25 of these seeds selected at random.

Calculate the expected number of seeds that germinate.

Find the probability that more than 7 of these seeds will germinate.

Find the probability that at least 9, but no more than 11 of the seeds germinate.

A bag contains 500 marbles in five colours: red, orange, yellow, green and blue.

There are $x$ red marbles, $3x$ orange marbles, $5x$ yellow marbles, $7x$ green marbles and $9x$ blue marbles.

Find the value of $x$.

Maeve selects a marble at random, records its colour and then returns it to the bag. Maeve follows this process 56 times.

Find the probability that none of the selected marbles are red.

Find the probability that at least a quarter of the selected marbles are red or blue.

Maeve decides not to return the marbles to the bag after she records their colours. State, with a reason, whether this would change the answers to parts (b) and (c).

A biased die is thrown 44 times and the expected number of times it will land on a six is 33.

The die is thrown another eight times.

Find the expected number of sixes seen from the eight throws.

Calculate the variance of the number of sixes seen from the eight throws. Give your answer as a fraction.

Find the probability that exactly four sixes are seen from the eight throws. Give your answer as a fraction.

When a pizza store receives a delivery order via their app the customer is told a predicted delivery time. The probability that the pizza store delivers the order before the predicted delivery time is $\frac{9}{20}$. During a given week, the pizza store makes 160 deliveries.

Find the expected number of deliveries delivered before the predicted time during that week.

Calculate the probability that they deliver over 100 orders before the predicted delivery time during that week. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathrm{\mathbb{Z}}$.

The manager, Ciro, decides that it is more important that the store delivers within 5 minutes of the predicted delivery time, either before or after it, as he believes that a delivery being too early is equally as inconvenient as being too late. He finds that the constant probability for this is $\frac{3}{4}$.

Calculate the probability that over 100 orders were delivered within 5 minutes of the predicted time during the week with 160 deliveries.

John has a collection of 12 suit jackets, of which 5 are tailored to fit him perfectly. On any given day, there is a 90% chance that John will wear a suit jacket and if he does, he chooses the suit jacket randomly.

Find the probability that on a given day John wears a tailored suit jacket.

In a year John works 260 days.

Find the expected number of days in a year that John wears a suit jacket.

Find the probability that John wears a non-tailored suit jacket to work at least 150 days in a year.

Greg sells A4 paintings for $14.75 each at a market. The probability that someone at the market buys a painting from his stall is 0.06. On a given day there are 250 people at the market.

Find the expected number of people that will buy a painting from Greg’s stall.

Find the probability that 20 or more people buy a painting from Greg’s stall.

Find the probability that Greg earns more than $118.

In a game, players select pieces of string at random from a box containing a large number of pieces of string of different lengths. The length, in cm, of a randomly chosen piece of string follows a continuous uniform distribution on the interval [4, 9].

A piece of string is selected at random from the box.

Find the probability that the piece of string is shorter than 7.5 cm.

A player selects five pieces of string. They win a chocolate bar if the longest of the five pieces is more than 7.5 cm long.

Find the probability that the player wins a chocolate bar.

A player selects eight pieces of string. They win a box of chocolates if five or more of the pieces of string are shorter than 5.2 cm.

Find the probability that the player wins a box of chocolates.

Zara is a gymnast.  It is known that she has a 20% chance of making a mistake in any given routine.

Zara performs ten routines in a competition.

(i) Find the expected number of routines in which Zara will make a mistake.

(ii) Find the standard deviation of the number of routines in which Zara makes a mistake.

Find the probability that Zara makes a mistake in:

(i) none of her routines,

(ii) exactly two of her routines,

(iii) no more than two of her routines.

Given that Zara makes a mistake in at least 2 of her routines, find the probability that she makes a mistake in exactly 3 of her routines.

Find the probability that the number of routines in which Zara makes a mistake is less than one standard deviation away from the mean.

A mall has a “clothing” section for clothes and an “other” section for everything else. The probability that a randomly observed customer goes to the clothing section is 0.78, and the probability that a randomly observed customer goes to the other section is 0.42. Assume that each customer goes to at least one section.

Find the probability that a randomly observed customer

(i) goes to both sections,

(ii) only goes to the clothing section.

On a given day the mall has 341 customers.

On this day, find

(i) the expected number of customers that go to the other section,

(ii) the probability that at least 250 customers visit the clothing section.

In London, England, the probability that a football player, who plays club football, is left-footed is 0.24. One particular squad, Raiders FC, has 25 players.

Find the probability that Raiders FC has exactly the expected number of left-footed players.

Find the probability that there are no left-footed players on the Raiders FC squad.

Find the probability that there are more right-footed players than left-footed players on the Raiders FC squad.

Suggest one assumption that has been made in this question.

Each of the 360 students at a school is put into one of three groups at random. The probability of a student being put into group A is $\frac{1}{12}$. A student is twice as likely to be put into group B as into group C.

Find the expected number of students put into each group.

20 students are chosen at random to form a sample.

Find the probability that at least four of the students are in group A.

Find the probability that less than 15 of the students are in groups A or B.

The headteacher of the school now wants exactly $\frac{1}{12}$ of the students to be in group A. Explain why the number of group A students in the sample can not be modelled by a binomial distribution.

A swimming training session for Ben includes swimming 12 lengths of 100 m. Ben can swim 100 m in less than 60 s with a constant probability of $\frac{5}{6}$. Let $X$ be the number of times Ben completes the 100 m length in less than 60 s during a training session.

Calculate the probability that Ben completes the 100 m in less than 60 s for all the 12 lengths.

Find the probability that Ben completes the 100 m in less than 60 s at least twice but no more than 9 times during a training session.

Ben’s younger brother, Zack, is doing the session with Ben and can complete the 100 m in less than 60 s with a constant probability of  $\frac{2}{5}$.

Find the probability that both Ben and Zack complete the 100 m in less than 60 s more than four times during a training session.

Suggest a reason why, in practice, the probability that Ben completes a length in less than 60 s might not be constant.

At a university, it is known from previous cohorts that the probability that a student is taller than 1.8 m is $\frac{3}{10}$ and the probability that a student weighs less than 80 kg is $\frac{9}{10}$. The university currently has 820 students.

Find the mean and variance for the number of students that

(i) are 1.8 m or shorter.

(ii) weigh 80 kg or more.

Assuming that a student’s height and weight are independent, find the probability that 250 students or more are taller than 1.8 m and weigh less than 80 kg.

Comment on the assumption that a student’s height and weight are independent.

In a restaurant, it is known that there is an 8 out of 15 chance that a guest will ask for water with their meal.

A random sample of 25 guests is taken.

Find the probability that

(i) over half of the guests ask for water,

(ii) between 24% and 92%, inclusive, of the guests ask for water.

A second random sample of 100 guests is selected. Let $X$ be the random variable that represents the number of guests that ask for water.

Find the smallest value of $k$ such that

$\mathrm{P}(X\le k)\ge 0.95$

A box contains many marbles, $\frac{2}{5}$ of which are blue. The rest are white. Marbles are picked at random from the box.

From drawing three marbles, with replacement, the probability of getting $x$ blue marbles is $\frac{36}{125}$. Find the value of $x$.

A multiple-choice test consists of 50 questions. Each question has 5 options of which only one is correct.

Henry takes the exam and randomly chooses one of the five answers for each question.

Find the expected number of questions Henry answers incorrectly.

Henry must get at least 25 of the questions correct to pass the test.

Find the probability that Henry passes the test. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathrm{\mathbb{Z}}$.

As Henry is going through the test, he realizes he is certain that he knows the answer to 12 of the questions.

Find the probability that Henry passes, given that he gets these 12 questions all correct.

Scott works as a real estate agent. The probability that Scott sells a house to a potential buyer over the age of 30 is 0.11, and the probability that he sells a house to a potential buyer aged 30 or under is 0.04.

In a given week, Scott meets 32 potential buyers, 17 of whom are over 30 and the rest are 30 or under. Assume that each potential buyer acts independently.

Find the expected number of houses that Scott sells in a week.

Find the probability that Scott sells at least one house in a week.

A multiple-choice test consists of 10 questions, where only one option is correct; six questions have three possible options and the other four questions have four possible options. Isaac takes the test and randomly selects an answer for each question.

Find the expected number of questions Isaac answers correctly.

Find the probability that Isaac gets exactly two answers correct.