Binomial & Geometric Distribution (Cambridge (CIE) AS Maths: Probability & Statistics 1): Exam Questions

A fair coin is tossed 20 times. The random variable $X$ is the number of times the coin lands heads up.

State, in context, two assumptions needed for $X$ to be modelled by a binomial distribution.

Find the probability that the coin lands heads up 15 times.

For a population of jellyfish in a certain area of the ocean, the probability that any given jellyfish contains microplastic particles in its body is 0.95.

State, in context, two assumptions needed for the number of jellyfish containing microplastic particles, in a sample of size $n$, to be modelled by a binomial distribution.

A random sample of 40 jellyfish from this area is selected.

Find the probability that

(i) exactly 38 of these jellyfish contain microplastic particles,

(ii) all 40 of these jellyfish contain microplastic particles.

On any day in a 5-day working week, the probability that Yussuf catches a taxi to work is 0.6, independently of other days.

Find the expected number of days in a 5-day working week on which Yussuf catches a taxi to work.

Find the probability that, in a randomly chosen 5-day working week, Yussuf never catches a taxi to work.

Find the probability that, in a randomly chosen 5-day working week, Yussuf catches a taxi to work at most once.

Derek watches the cars that pass his window. The probability that any passing car is blue is 0.25, independently of other cars.

Find the probability that the first blue car Derek sees is the 8th car to pass his window.

Find the probability that Derek sees a blue car within the first 8 cars that pass his window.

Find the expected number of cars that pass Derek's window, up to and including the first blue car.

Ravi throws a ball at a target until he hits it for the first time. On each throw, the probability that he hits the target is $p$, independently of other throws. The random variable $X$ is the number of throws Ravi takes, up to and including his first hit.

Given that $\text{E}\left(X\right)=\frac{4}{3}$, find the value of $p$.

Find

(i) $\text{P}(X>6)$,

(ii) $\text{P}(X\le 5)$.

On any school day, the probability that a particular bus arrives late is 0.4, independently of other days. The random variable $Y$ is the number of days, out of $n$ school days, on which the bus arrives late.

Given that $\text{E}\left(Y\right)=12$, find the value of $n$.

Find

(i) $\text{P}(8<Y<12)$,

(ii) $\text{Var}\left(Y\right)$.

Giovanni rolls a biased die for which the probability of landing on a two on any roll is 0.25, independently of other rolls.

Find the probability that Giovanni rolls a two for the first time on his 11th roll.

Giovanni rolls the die 10 times.

Find the probability that the die lands on a two exactly 4 times.

Cans of a particular brand of soft drink are labelled as containing 330 ml. The probability that a randomly chosen can contains less than 320 ml is 0.0296.

Tilly buys a pack of 24 of these cans, which may be regarded as a random sample. The random variable $L$ is the number of cans in the pack that contain less than 320 ml.

State the distribution of $L$, giving the values of its parameters.

Find the probability that

(i) none of the cans,

(ii) exactly two of the cans,

(iii) at least two of the cans

in Tilly's pack contain less than 320 ml.

In an experiment, the number of specimens testing positive for a certain characteristic is modelled by the random variable $X~\text{B}(20,0.35)$. Find the probability of

(i) fewer than 3

(ii) no more than 3

(iii) at least 3

(iv) at most 3

(v) more than 3

of the specimens testing positive for the characteristic.

In the town of Wooster, the probability that a randomly chosen resident prefers the locally made Woostershire brand of sauce is 0.9. The other residents prefer another brand.

A random sample of 30 residents is selected. The random variable $X$ is the number of these residents who prefer Woostershire brand sauce.

State, in context, two assumptions needed for $X$ to be modelled by a binomial distribution.

Find the probability that

(i) more than 90% of the residents in the sample prefer Woostershire brand sauce,

(ii) none of the residents in the sample prefer the other brand.

Find

(i) $\text{E}\left(X\right)$,

(ii) $\text{Var}\left(X\right)$.

A snowboarder is practising a trick called the Poptart. On each attempt, the probability that the snowboarder completes the trick successfully is 0.25. The snowboarder makes 12 attempts.

(i) State, in context, one assumption needed for a binomial distribution to be a suitable model for the number of successful attempts.

(ii) Give a reason, in context, why this assumption may not hold.

Find the probability that the snowboarder

(i) completes the trick successfully more than 2 times in the 12 attempts,

(ii) does not complete the trick successfully on any of the 12 attempts.

Gregg is a baker. When he cracks open an egg, the probability that he breaks the yolk is 0.05.

State, in context, two assumptions needed for the number of eggs Gregg cracks open, up to and including the first one in which he breaks the yolk, to be modelled by a geometric distribution.

Find the probability that

(i) Gregg breaks a yolk for the first time on the 10th egg he cracks open,

(ii) Gregg breaks a yolk within the first 4 eggs he cracks open.

To join a superhero club, Bucky must pass a superhero examination. Bucky takes the examination repeatedly until he passes. The probability that Bucky passes the examination on any attempt is 0.16, independently of all previous attempts.

Find the probability that

(i) Bucky passes on his 5th attempt,

(ii) Bucky takes at least 5 attempts to pass,

(iii) Bucky passes within his first 10 attempts.

Find the expected number of attempts Bucky needs to pass the examination.

State one criticism of this model, in context.

The probability that an energy-saving light bulb made by a certain company has a lifetime of less than 5000 hours is 0.1. A random sample of 30 of these light bulbs is taken.

State, in context, two assumptions needed for the number of light bulbs in the sample with a lifetime of less than 5000 hours to be modelled by a binomial distribution.

Find the probability that, in the sample of 30 light bulbs,

(i) exactly one light bulb has a lifetime of less than 5000 hours,

(ii) no more than three light bulbs have a lifetime of less than 5000 hours.

Abner, an American baseball fan, has just moved to a town in which it is known that 85% of the residents are familiar with the rules of the game.

Abner takes a random sample of 40 residents of the town. Find the probability that

(i) fewer than 38

(ii) no more than 38

(iii) more than 38

(iv) at most 38 but at least 36

of the residents in Abner's sample are familiar with the rules of baseball.

Farmer Kate has a herd of 50 alpacas. From previous testing of the herd, she knows that each alpaca has a probability of 0.95 of testing negative for tuberculosis (TB). She chooses a random sample of 8 alpacas from the herd and tests them for TB.

$N$ is the number of alpacas in the sample that test negative for TB, and $T$ is the number of alpacas in the sample that test positive for TB.

(i) State the distribution of $N$, giving the values of its parameters.

(ii) State the distribution of $T$, giving the values of its parameters.

Find the probability that the number of alpacas in the sample that test positive for TB is

(i) zero,

(ii) more than 2.

A fair six-sided die is rolled 24 times. The random variable $X$ is the number of times the die lands on a 4.

State, in context, two assumptions needed for $X$ to be modelled by a binomial distribution.

Find the probability that the die lands on a 4 exactly four times.

In a certain area of woodland, the probability that any given squirrel was born in that area is 0.92. Squirrels born in that area are described by researchers as "local".

State, in context, two assumptions needed for the number of local squirrels, in a sample of size $n$, to be modelled by a binomial distribution.

A random sample of 50 squirrels from this area is selected.

Find the probability that

(i) exactly 45 of these squirrels are local,

(ii) all but one of these squirrels are local.

The probability that a seed of a certain variety fails to germinate is 0.15, independently of other seeds. A random sample of 40 of these seeds is planted. The random variable $X$ is the number of these seeds that fail to germinate.

Find

(i) $\text{E}\left(X\right)$,

(ii) $\text{Var}\left(X\right)$.

Find the probability that

(i) exactly 5 of the seeds fail to germinate,

(ii) at least 3 of the seeds fail to germinate.

Given that at least 3 of the seeds fail to germinate, find the probability that exactly 5 of them fail to germinate.

The random variable $X~\text{Geo}\left(p\right)$.

(i) Find the value of $p$ when $\text{P}(X\ge 3)=0.09$.

(ii) Find the possible values of $p$ when $\text{P}(X=2)=0.2275$.

The random variable $Y~\text{B}(5,p)$.

(i) Find the value of $p$ when $\text{E}\left(Y\right)=3$.

(ii) Find the value of $p$ when $\text{P}(Y=5)=0.32768$.

(iii) Find the possible values of $p$ when $\text{Var}\left(Y\right)=1.05$.

Guglielma rolls a biased die for which the probability of landing on a 5 on any roll is $\frac{2}{11}$, independently of other rolls.

Find the probability that Guglielma rolls a 5 for the first time on her 21st roll.

Guglielma rolls the die 20 times.

Find the probability that

(i) the die lands on a 5 exactly 4 times,

(ii) the die lands on a 5 exactly 4 times, but the final 5 does not occur on the 20th roll.

Bars of a particular brand of chocolate are labelled as weighing 300 g. The probability that a randomly chosen bar weighs between 297 g and 303 g is 0.9596. For bars outside this range, the probability that a bar weighs less than 297 g is equal to the probability that it weighs more than 303 g.

Millie buys 25 of these bars, which may be regarded as a random sample. The random variable $U$ is the number of these bars that weigh less than 297 g.

State the distribution of $U$, giving the values of its parameters.

Millie plans to hand out the bars at a meeting attended by 24 people. Bars weighing less than 297 g cannot be handed out.

Find the probability that there will be enough bars to hand out to

(i) all 24 people,

(ii) all but one of the 24 people, but not all of them.

Two fair six-sided dice are rolled and the numbers showing are added together. This is done 18 times. The random variable $X$ is the number of times the sum is not equal to 7 or 11.

(i) State the distribution of $X$, giving the values of its parameters.

(ii) State, in context, two assumptions needed for $X$ to be modelled by this distribution.

Find the probability that the sum is not equal to 7 or 11 on exactly 14 of the 18 occasions.

In a certain region, the probability that any given female mosquito carries the malaria parasite is 0.8.

State, in context, two assumptions needed for the number of female mosquitoes carrying the parasite, in a sample of $n$ female mosquitoes, to be modelled by a binomial distribution.

Male mosquitoes cannot transmit the malaria parasite to a human. A female mosquito can transmit the parasite to a human only if she carries it. In this region, 50% of the mosquitoes are male.

Find the probability that, in a random sample of six mosquitoes, none of them is able to transmit the parasite to a human. Give your answer as an exact value.

Zara is a gymnast. The probability that she makes a mistake in any given routine is 0.2, independently of other routines. Zara performs 10 routines in a competition. The random variable $X$ is the number of these routines in which Zara makes a mistake.

Find

(i) $\text{E}\left(X\right)$,

(ii) the standard deviation of $X$.

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 $X$ is less than one standard deviation away from the mean.

David stands outside a library and asks people who walk past him to answer some market research questions. The probability that a person stops is 0.05, independently of other people.

Find the expected number of people who walk past without stopping before someone stops.

Find the probability that

(i) the first 20 people walk past David without stopping,

(ii) at least one of the first 100 people who walk past stops.

Given that the first 20 people walk past without stopping, find the probability that at least one of the first 100 people who walk past stops.

Find the probability that the 10th person and the 20th person to walk past are the first two people to stop.

Find the least number of people who must walk past so that the probability that at least one of them stops is at least 0.999.

Anand and Beatrix each throw a dart at a target repeatedly until they hit it for the first time. On each throw, the probability that the thrower hits the target is $p$, independently of all other throws by either of them.

(i) Show that the probability that Anand and Beatrix make a total of 4 throws between them can be written as $k{p}^2{\left(1-p\right)}^2$, where $k$ is a constant to be found.

(ii) Given that the probability that they make a total of 4 throws is 0.1875, find the value of $p$.

The probability that Chen wins a game of chess against a computer is 0.2, independently of other games. Chen plays $n$ games. The random variable $Y$ is the number of these games that Chen wins.

Find the largest value of $n$ for which the probability that Chen wins at least one game is less than 0.9.

Maifreda rolls a biased die with faces numbered 1 to 6. The probability that the die lands on a prime number is $\frac{1}{2}$ and the probability that it lands on a square number is $\frac{5}{16}$.

Find the expected number of rolls needed to obtain a 6 for the first time.

Maifreda rolls the die 20 times.

Find the probability that

(i) the die lands on a 6 exactly four times,

(ii) the die lands on a 6 exactly four times, and all four 6s occur within the first $k$ rolls, where $4\le k\le 20$. Give your answer in terms of $k$.