Working with Distributions (Cambridge (CIE) A Level Maths: Probability & Statistics 1): Exam Questions

At a large college, 75% of the students walk to college. A random sample of 8 students from the college is chosen.

Find the probability that at least 7 of these 8 students walk to college.

A second random sample, of 75 students from the college, is chosen.

Use an approximation to find the probability that between 55 and 57 inclusive of these 75 students walk to college.

Justify the use of your approximation in part (b).

It is known that 40% of the seeds of a certain variety germinate. A random sample of 500 of these seeds is sown.

Use an approximation to find the probability that the number of these seeds that germinate is between 189 and 211 inclusive.

Use the same approximation to find the probability that more than 220 of these seeds germinate.

Justify the use of your approximation in part (a).

Using your approximation, find the greatest integer value of $w$ such that the probability that at most $w$ of these seeds germinate is less than 0.1.

In a computer game, the probability that a player wins any round is $\frac{9}{19}$, independently of all other rounds.

For a random sample of 30 rounds, find the probability that the player wins exactly 14 rounds.

A random sample of 1000 rounds is now chosen.

Use an approximation to find the probability that the player wins at least 500 of these rounds.

Justify the use of your approximation in part (b).

As part of a marketing promotion, 47% of packets of a particular brand of crisps contain a toy as a prize. A random sample of 100 packets is taken.

Find the probability that exactly 49 of the packets contain a prize. Give your answer correct to 4 significant figures.

Write down the normal distribution that could be used to approximate the distribution of the number of the 100 packets that contain a prize.

Use your normal approximation to find an approximate value for the probability that exactly 49 of the packets contain a prize.

The weights, in grams, of the chocolate bars produced by a certain manufacturer are normally distributed with mean 200 and standard deviation 17.5.

Find the probability that a randomly chosen chocolate bar weighs less than 195 grams.

Hilda chooses chocolate bars at random, one at a time.

Find the probability that the 8th chocolate bar that she chooses is the first one that weighs less than 195 grams.

A random sample of 12 of these chocolate bars is chosen.

Find the probability that exactly 2 of these chocolate bars weigh less than 195 grams.

A machine produces plastic counters. Each counter is red with probability $\frac{6}{13}$, black with probability $\frac{6}{13}$, and otherwise green, independently of all other counters.

A random sample of 50 counters is chosen. Find the probability that exactly 45 of these counters are either red or black.

In a separate experiment, a random sample of 1000 counters is chosen.

Use a suitable approximation to find the probability that at least 75 but fewer than 80 of these counters are green.

Justify the use of your approximation in part (b).

At a large supermarket, 60% of customers pay by card. A random sample of 1200 customers is chosen.

Use a suitable approximation to find the probability that the number of these customers who pay by card is more than 700 and at most 730.

Use your approximation to find the probability that at least 719 of these customers pay by card.

Justify the use of your approximation in part (a).

Using your approximation, find the greatest integer value of $k$ such that the probability that the number of these customers who pay by card differs from 720 by less than $k$ is less than 0.5.

The probability that a seed of a certain type produces a red flower is $\frac{18}{37}$.

A random sample of 36 seeds of this type is chosen. Find the probability that exactly 18 of these seeds produce red flowers.

In a separate experiment, a random sample of 1000 seeds of this type is chosen.

Use a suitable approximation to find the probability that fewer than 482 or more than 491 of these seeds produce red flowers.

Justify the use of your approximation in part (b).

The weights of the feed in sacks of partridge feed produced by a certain manufacturer are normally distributed with mean 20 kg and standard deviation 0.1 kg.

Find the probability that a randomly chosen sack contains less than 19.75 kg of feed.

Roger buys 10 sacks of this feed, chosen at random.

Find the probability that the weight of feed in every one of these 10 sacks differs from the mean weight by less than 0.25 kg.

Find the probability that, for at least two of these 10 sacks, the weight of feed differs from the mean weight by 0.25 kg or more.

A farm grows two varieties of cucumber, A and B.

The lengths of cucumbers of variety A are normally distributed with mean 30.0 cm and standard deviation 2.5 cm.

In a random sample of 400 cucumbers of variety A, how many would you expect to have a length between 27.0 cm and 33.5 cm?

The lengths of cucumbers of variety B are normally distributed with mean 28.0 cm and standard deviation $\sigma$ cm. It is found that 15% of cucumbers of variety B have a length less than 25.0 cm.

Find the value of $\sigma$.

A random sample of 90 cucumbers of variety B is chosen.

Use a suitable approximation to find the probability that fewer than 10 of these 90 cucumbers have a length less than 25.0 cm.

The lengths, in metres, of metal rods produced by a machine are normally distributed with mean $\mu$ and standard deviation $\sigma$. The interquartile range of the lengths is 0.01712 m.

Find the value of $\sigma$.

Find the probability that a randomly chosen rod is shorter than the mean length by 1 cm or more.

Twenty of the rods are chosen at random.

Find the probability that at most two of these rods are shorter than the mean length by 1 cm or more.

In a game, Ravi throws five fair coins at the same time. He wins a point if he obtains at least 4 heads.

Show that the probability that Ravi wins a point on any throw of the five coins is $\frac{3}{16}$.

Find the probability that Ravi wins a point for the first time on his 5th throw of the five coins.

Ravi throws the five coins on 120 occasions.

Use an approximation to find the probability that he wins a point on more than 25 of these occasions.

Justify the use of your approximation in part (c).

In a large town, 40% of adults read the local newspaper. A random sample of 980 adults from the town is chosen.

Use a suitable approximation to find the probability that more than 386 but fewer than 398 of these adults read the local newspaper.

Use your approximation to find the probability that at least 400 of these adults read the local newspaper.

Justify the use of your approximation in part (a).

Using your approximation, find the smallest integer value of $k$ such that the probability that more than $k$ but fewer than $784-k$ of these adults read the local newspaper is less than 0.5.