2.04c Calculate binomial probabilities

In a particular country, 8% of the population has blue eyes. A random sample of 20 people is selected from this population. Find the probability that exactly two of these people have blue eyes. [2]

The random variable \(X\) has the binomial distribution B(16, 0·3). Showing your calculation, find \(P(X = 7)\). [2]

Naomi produces oak tabletops, each of area 4·8 m². Defects in the oak tabletops occur randomly at a rate of 0·25 per m².

1. Find the probability that a randomly chosen tabletop will contain at most 2 defects. [3]
2. Find the probability that, in a random sample of 7 tabletops, exactly 4 will contain at most 2 defects each. [3]

At a fairground, Kirsty throws \(n\) balls in order to try to knock coconuts off their stands. Any coconuts she knocks off are replaced before she throws again. Kirsty counts the number of coconuts she successfully knocks off their stands. On average, she knocks off a coconut with 20\% of her throws.

1. What assumptions are needed in order to model this situation with a binomial distribution? Explain whether these assumptions are reasonable. [2]

Kirsty uses a spreadsheet to produce the following diagrams, showing the probability distributions of the number of coconuts knocked off their stands for different values of \(n\). \includegraphics{figure_3}

1. Describe two ways in which the distribution changes as \(n\) increases. [2]

The weights of sacks of potatoes are normally distributed. It is known that one in five sacks weigh more than 6kg and three in five sacks weigh more than 5.5kg.

1. Find the mean and standard deviation of the weights of potato sacks. [5]
2. The sacks are put into crates, with twelve sacks going into each crate. What is the probability that a given crate contains two or more sacks that weigh more than 6kg? You must explain your reasoning clearly in this question. [4]

Every time a spinner is spun, the probability that it shows the number 4 is 0.2, independently of all other spins.

1. A pupil spins the spinner repeatedly until it shows the number 4. Find the mean of the number of spins required. [2]
2. Calculate the probability that the number of spins required is between 3 and 10 inclusive. [2]
3. Each pupil in a class of 30 spins the spinner until it shows the number 4. Out of the 30 pupils, the number of pupils who require at least 10 spins is denoted by \(X\). Determine the variance of \(X\). [4]

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.

1. 1. Find the mean number of times he falls off in a day. [1 mark]
   2. Find the variance of the number of times he falls off in a day. [1 mark]
2. 1. Find the probability that, on a particular day, he falls off exactly 10 times. [2 marks]
   2. Find the probability that, on a particular day, he falls off 5 or more times. [3 marks]
3. Patrick has 30 attempts to perform the trick on each of 5 consecutive days.
   1. Calculate the probability that he will fall off his skateboard at least 5 times on each of the 5 days. [2 marks]
   2. Explain why it may be unrealistic to use the same value of 0.2 for the probability of falling off for all 5 days. [1 mark]

The random variable \(X\) represents the weight in kg of a randomly selected male dog of a particular breed. \(X\) is Normally distributed with mean 30.7 and standard deviation 3.5.

1. Find the 90th percentile for the weights of these dogs. [2]
2. Five of these dogs are chosen at random. Find the probability that exactly four of them weighs at least 30 kg. [3]

The weights of females of the same breed of dog are Normally distributed with mean 26.8 kg.

1. Given that 5% of female dogs of this breed weigh more than 30 kg, find the standard deviation of their weights. [3]
2. Sketch the distributions of the weights of male and female dogs of this breed on a single diagram. [3]

A retail bakery makes cherry muffins where, due to the production process, 15% of muffins contain a lower than expected quantity of cherries. The bakery sells these muffins in boxes of 20.

1. State a suitable distribution to model the number of muffins with a lower than expected quantity of cherries in a box, giving the value(s) of any parameter(s). State any assumptions needed for your model to be valid. [4]
2. Using your model from part (a), find the probability that a randomly selected box contains:
   1. exactly 3 muffins with a lower than expected quantity of cherries, [2]
   2. at least 5 muffins with a lower than expected quantity of cherries. [2]
3. The bakery sells 25 boxes of muffins in one day. Find the probability that fewer than 4 of these boxes contain exactly 3 muffins with a lower than expected quantity of cherries. [3]

In a factory, computer chips are produced in large batches. A quality control procedure is used for each batch which requires a random sample of 8 chips to be tested. If no faulty chip is found, the batch is accepted. If two or more are faulty, the batch is rejected. If one is faulty, a further sample of 4 is selected and the batch is accepted if none of these is faulty. The probability of any chip being faulty is \(q\).

1. Show that the probability of accepting a batch is \(p^8(1 + 8p^3 - 8p^4)\), where \(p = 1 - q\). [6]
2. Find the expected number of chips sampled per batch, giving your answer in terms of \(p\). Hence show that when \(p = 0.75\), the expected number of chips sampled per batch is approximately 9. [6]

In a certain country 40% of the population have brown eyes. A random sample of 20 people is chosen from that population.

1. Find the expected number of people in the sample who have brown eyes. [1]
2. Find the probability that there are exactly 8 people with brown eyes in the sample. [3]
3. Find the probability that there are at least 8 people with brown eyes in the sample. [2]

In a certain country 40\% of the population have brown eyes. A random sample of 20 people is chosen from that population.

1. Find the expected number of people in the sample who have brown eyes. [1]
2. Find the probability that there are exactly 8 people with brown eyes in the sample. [3]
3. Find the probability that there are at least 8 people with brown eyes in the sample. [2]

A certain type of sweet is made in a variety of colours. \(20\%\) of the sweets made are blue. Sweets of the various colours are thoroughly mixed before being put into packets.

1. In a packet that contains 10 sweets, find the probability that the packet contains
   1. at most 3 blue sweets, [1]
   2. exactly 3 blue sweets, [2]
   3. at least 1 blue sweet. [2]
2. What is the smallest number of sweets that a packet should contain in order to be at least \(95\%\) certain of having at least 1 blue sweet? [3]

A survey into left-handedness found that 13% of the population of the world are left-handed.

1. State the assumptions necessary for it to be appropriate to model the number of left-handed children in a class of 20 children using the binomial distribution B(20, 0.13). [2]
2. Assuming that this binomial model is appropriate, calculate the probability that fewer than 13% of the 20 children are left-handed. [4]

A survey into left-handedness found that 13% of the population of the world are left-handed.

1. State the assumptions necessary for it to be appropriate to model the number of left-handed children in a class of 20 children using the binomial distribution B(20, 0.13). [2]
2. Assuming that this binomial model is appropriate, calculate the probability that fewer than 13% of the 20 children are left-handed. [4]