Binomial Distributions (DP IB Maths: AA SL)

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.

At a builders’ convention there is 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.

A biased dice is rolled 24 times. The probability that the dice lands on a prime number is   . The dice 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 dice lands on.

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

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

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

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.

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 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.

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  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 .

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

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  inside. A contestant randomly selects a box and wins the amount that is inside.

Find the expected payoff from playing the game.

Suppose there are 28 contestants.

Find the expected number of players that win

$10

$100

Find the probability that

exactly four contestants win $10 000

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.

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

In a given week, Scott interacts with 32 potential buyers, 17 of which are over 30 and the rest are 30 or under.

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

Find the probability 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.

A fair coin is tossed seven times.

Find the probability of getting exactly five tails. Give your answer as a fraction.

Find the probability of getting exactly three heads. Give your answer as a fraction.

On any given day, the probability that Mark rides his bike to school is   .  

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.

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.

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   . 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 , where  and .

The manager, Ciro, decides that it is more important that they deliver the delivery within 5 minutes of the predicted delivery time, either 5 minutes before or 5 minutes after the predicted time. As he believes a delivery being too early is equally as inconvenient as being too late. He finds that the constant probability for this measure is 

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

At a university, it is known from previous cohorts that the probability that a student is taller than 1.8 m is   and the probability that a student weighs less than 80 kg is   . The university currently has 820 students.

Find the mean and variance for the number of students that

are 1.8 m or shorter.

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.

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  . Let 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  .

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 as to why the probability for Ben and Zack to complete the 100 m in less than 60 s is not constant.

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 Henry passes the test. Give your answer in the form , where and .

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 Henry passes, given he gets these 12 questions all correct.

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

goes to both sections,

On a given day the mall has 341 customers.

On this day, find

the expected number of customers that go to the other section,

360 students at a school are divided into three groups. The probability of a student being put into group A is . A student is twice as likely to be put into group B than 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 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 bag of 500 marbles are divided into five colours: red, orange, yellow, green and blue.

There are  red marbles,  orange marbles,  yellow marbles,  green marbles and  blue marbles.

Find the value of .

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 part (b) and (c).

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 are sampled.

Find the probability that

over half of the guests ask for water,

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

Find the smallest value of such that

In a game, players select pieces of string 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 has uniform distribution over 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.

If a player selects five pieces of string, then they win a chocolate bar if the length of the longest piece of string is more than 7.5 cm.

A player selects five pieces of string, find the probability of winning a chocolate bar.

If a player selects eight pieces of string, they then win a box of chocolates if five or more of the pieces of string are shorter than 5.2 cm

A player selects eight pieces of string, find the probability of winning a box of chocolates.

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

The dice 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.

A box contains many marbles,  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  blue marbles is   . Find the value of .