Binomial distribution

The proportion of diners at a restaurant who are vegan is 20%

The number of diners at the restaurant who are vegan can be modelled by a binomial distribution.

In a random sample of 25 diners:

find the probability that no diner is a vegan.

find the probability that fewer than 6 diners are vegan.

find the probability that at least 9 diners are vegan.

State one assumption necessary for the binomial distribution to be valid in the context of this question.

It is known that, on average, 40% of the drivers who take their driving test at a local test centre pass their driving test.

Each day 32 drivers take their driving test at this centre.

The number of drivers who pass their test on a particular day can be modelled by the distribution B (32, 0.4)

State one assumption, in context, required for this distribution to be used.

Find the probability that exactly 7 of the drivers on a particular day pass their test.

Find the probability that, at most, 16 of the drivers on a particular day pass their test.

Find the probability that more than 12 of the drivers on a particular day pass their test.

Find the mean number of drivers per day who pass their test.

Find the standard deviation of the number of drivers per day who pass their test.

A customer service centre records every call they receive.

It is found that 30% of all calls made to this centre are complaints.

A sample of 20 calls is selected.

The number of calls in the sample which are complaints is denoted by the random variable X.

State two assumptions necessary for X to be modelled by a binomial distribution.

Assume that X can be modelled by a binomial distribution.

Find P(X 1)

Find P(X 4)

Find P(X 10)

In a random sample of 10 calls to a school, the number of calls which are complaints, Y, may be modelled by a binomial distribution

The standard deviation of Y is 1.5

Calculate the possible values of p.

The random variable X is such that X B(n, p)

The mean value of X is 225

The variance of X is 144

Find p.

Circle your answer.

0.36 0.6 0.64 0.8

James is playing a mathematical game on his computer.

The probability that he wins is 0.6

As part of an online tournament, James plays the game 10 times.

Let Y be the number of games that James wins.

State two assumptions, in context, for Y to be modelled as B (10, 0.6)

Find P(Y 4)

Find P(Y 4)

After practising the game, James claims that he has increased his probability of winning the game.

In a random sample of 15 subsequent games, he wins 12 of them.

Test a 5% significance level whether James’s claim is correct.

Tiana is a quality controller in a clothes factory. She checks for four possible types of defects in shirts.

Of the shirts with defects, the proportion of each type of defect is as shown in the table below.

Shirts with defects are packed in boxes of 30 at random.

Find the probability that:

a box contains exactly 5 shirts with a colour defect

a box contains fewer than 15 shirts with a sewing defect

a box contains at least 20 shirts which do not have a fabric defect.

Tiana wants to investigate the proportion, p, of defective shirts with a fabric defect.

She wishes to test the hypotheses

She takes a random sample of 60 shirts with a defect and finds that x of them have a fabric defect.

Using a 5% level of significance, find the critical region for x.

In her sample she finds 13 shirts with a fabric defect.

Complete the test stating her conclusion in context.

Patrick is practising his skateboarding skills. On each day, he has 30 attempts at performing a difficult trick.

Every time he attempts the trick, there is a probability of 0.2 that he will fall off his skateboard.

Assume that the number of times he falls off on any given day may be modelled by a binomial distribution.

Find the mean number of times he falls off in a day.

Find the variance of the number of times he falls off in a day.

Find the probability that, on a particular day, he falls off exactly 10 times.

Find the probability that, on a particular day, he falls off 5 or more times.

Patrick has 30 attempts to perform the trick on each of 5 consecutive days.

Calculate the probability that he will fall off his skateboard at least 5 times on each of the 5 days.

Explain why it may be unrealistic to use the same value of 0.2 for the probability of falling off for all 5 days.

Abu visits his local hardware store to buy six light bulbs.

He knows that 15% of all bulbs at this store are faulty.

State a distribution which can be used to model the number of faulty bulbs he buys.

Find the probability that all of the bulbs he buys are faulty.

Find the probability that at least two of the bulbs he buys are faulty.

Find the mean of the distribution stated in part (a).

State two necessary assumptions in context so that the distribution stated in part (a) is valid.