2.04c Calculate binomial probabilities

5 The weights of apples sold by a store can be modelled by a normal distribution with mean 120 grams and standard deviation 24 grams. Apples weighing less than 90 grams are graded as 'small'; apples weighing more than 140 grams are graded as 'large'; the remainder are graded as 'medium'.

1. Show that the probability that an apple chosen at random is graded as medium is 0.692 , correct to 3 significant figures.
2. Four apples are chosen at random. Find the probability that at least two are graded as medium. [4]

2 Annan has designed a new logo for a sportswear company. A survey of a large number of customers found that \(42 \%\) of customers rated the logo as good.

1. A random sample of 10 customers is chosen. Find the probability that fewer than 8 of them rate the logo as good.
2. On another occasion, a random sample of \(n\) customers of the company is chosen. Find the smallest value of \(n\) for which the probability that at least one person rates the logo as good is greater than 0.995 .

7 The shortest time recorded by an athlete in a 400 m race is called their personal best (PB). The PBs of the athletes in a large athletics club are normally distributed with mean 49.2 seconds and standard deviation 2.8 seconds.

1. Find the probability that a randomly chosen athlete from this club has a PB between 46 and 53 seconds.
2. It is found that \(92 \%\) of athletes from this club have PBs of more than \(t\) seconds. Find the value of \(t\).
   Three athletes from the club are chosen at random.
3. Find the probability that exactly 2 have PBs of less than 46 seconds.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 In Quarendon, \(66 \%\) of households are satisfied with the speed of their wifi connection.

1. Find the probability that, out of 10 households chosen at random in Quarendon, at least 8 are satisfied with the speed of their wifi connection.
2. A random sample of 150 households in Quarendon is chosen. Use a suitable approximation to find the probability that more than 84 are satisfied with the speed of their wifi connection. [5]

7 A competition is taking place between two choirs, the Notes and the Classics. There is a large audience for the competition.

• \(30 \%\) of the audience are Notes supporters.
• \(45 \%\) of the audience are Classics supporters.
• The rest of the audience are not supporters of either of these choirs.
• No one in the audience supports both of these choirs.

1 In a certain town, 76\% of cars are fitted with satellite navigation equipment. A random sample of 11 cars from this town is chosen. Find the probability that fewer than 10 of these cars are fitted with this equipment.

7 The faces of a biased die are numbered \(1,2,3,4,5\) and 6 . The probabilities of throwing odd numbers are all the same. The probabilities of throwing even numbers are all the same. The probability of throwing an odd number is twice the probability of throwing an even number.

1. Find the probability of throwing a 3 . \includegraphics[max width=\textwidth, alt={}, center]{34ae4f06-d485-4138-82d8-902b70f08995-10_51_1563_495_331}
2. The die is thrown three times. Find the probability of throwing two 5 s and one 4 .
3. The die is thrown 100 times. Use an approximation to find the probability that an even number is thrown at most 37 times.

2 Arvind uses an ordinary fair 6-sided die to play a game. He believes he has a system to predict the score when the die is thrown. Before each throw of the die, he writes down what he thinks the score will be. He claims that he can write the correct score more often than he would if he were just guessing. His friend Laxmi tests his claim by asking him to write down the score before each of 15 throws of the die. Arvind writes the correct score on exactly 5 out of 15 throws. Test Arvind's claim at the \(10 \%\) significance level.

4 The independent random variables \(X\) and \(Y\) have distributions \(\operatorname { Po } ( 2 )\) and \(\mathrm { B } \left( 20 , \frac { 1 } { 4 } \right)\) respectively.

1. Find the mean and standard deviation of \(X - 3 Y\).
2. Find \(\mathrm { P } ( Y = 15 X )\).

2 Anton believes that \(10 \%\) of students at his college are left-handed. Aliya believes that this is an underestimate. She plans to carry out a hypothesis test of the null hypothesis \(p = 0.1\) against the alternative hypothesis \(p > 0.1\), where \(p\) is the actual proportion of students at the college that are left-handed. She chooses a random sample of 20 students from the college. She will reject the null hypothesis if at least 5 of these students are left-handed.

1. Explain what is meant by a Type I error in this context.
2. Find the probability of a Type I error in the test.
3. Given that the true value of \(p\) is 0.3 , find the probability of a Type II error in the test.

6 When a child completes an online exercise called a Mathlit, they might be awarded a medal. The publishers claim that the probability that a randomly chosen child who completes a Mathlit will be awarded a medal is \(\frac { 1 } { 3 }\). Asha wishes to test this claim. She decides that if she is awarded no medals while completing 10 Mathlits, she will conclude that the true probability is less than \(\frac { 1 } { 3 }\).

1. Use a binomial distribution to find the probability of a Type I error.
   The true probability of being awarded a medal is denoted by \(p\).
2. Given that the probability of a Type II error is 0.8926 , find the value of \(p\).

7 Every July, as part of a research project, Rita collects data about sightings of a particular kind of bird. Each day in July she notes whether she sees this kind of bird or not, and she records the number \(X\) of days on which she sees it. She models the distribution of \(X\) by \(\mathrm { B } ( 31 , p )\), where \(p\) is the probability of seeing this kind of bird on a randomly chosen day in July. Data from previous years suggests that \(p = 0.3\), but in 2022 Rita suspected that the value of \(p\) had been reduced. She decided to carry out a hypothesis test. In July 2022, she saw this kind of bird on 4 days.

1. Use the binomial distribution to test at the \(5 \%\) significance level whether Rita's suspicion is justified.
   In July 2023, she noted the value of \(X\) and carried out another test at the \(5 \%\) significance level using the same hypotheses.
2. Calculate the probability of a Type I error.
   Rita models the number of sightings, \(Y\), per year of a different, very rare, kind of bird by the distribution \(B ( 365,0.01 )\).
3. 1. Use a suitable approximating distribution to find \(\mathrm { P } ( Y = 4 )\).
   2. Justify your approximating distribution in this context.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 The random variable \(X\) has the distribution \(\mathrm { B } ( 4000,0.001 )\).

1. Use a suitable approximating distribution to find \(\mathrm { P } ( 2 \leqslant X < 5 )\).
2. Justify your approximating distribution in this case.

6 It is known that \(8 \%\) of adults in a certain town own a Chantor car. After an advertising campaign, a car dealer wishes to investigate whether this proportion has increased. He chooses a random sample of 25 adults from the town and notes how many of them own a Chantor car.

1. He finds that 4 of the 25 adults own a Chantor car. Carry out a hypothesis test at the 5\% significance level.
2. Explain which of the errors, Type I or Type II, might have been made in carrying out the test in part (a).
   Later, the car dealer takes another random sample of 25 adults from the town and carries out a similar hypothesis test at the 5\% significance level.
3. Find the probability of a Type I error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The following histogram illustrates the distribution of times, in minutes, that some students spent taking a shower. \includegraphics[max width=\textwidth, alt={}, center]{ec425eaf-8afc-4671-9ef3-ba2477b884ef-3_1031_1326_372_406}

1. Copy and complete the following frequency table for the data.

   Time \(( t\) minutes \()\)	\(2 < t \leqslant 4\)	\(4 < t \leqslant 6\)	\(6 < t \leqslant 7\)	\(7 < t \leqslant 8\)	\(8 < t \leqslant 10\)	\(10 < t \leqslant 16\)
   Frequency

2. Calculate an estimate of the mean time to take a shower.
3. Two of these students are chosen at random. Find the probability that exactly one takes between 7 and 10 minutes to take a shower.

7 The times spent by people visiting a certain dentist are independent and normally distributed with a mean of 8.2 minutes. \(79 \%\) of people who visit this dentist have visits lasting less than 10 minutes.

1. Find the standard deviation of the times spent by people visiting this dentist.
2. Find the probability that the time spent visiting this dentist by a randomly chosen person deviates from the mean by more than 1 minute.
3. Find the probability that, of 6 randomly chosen people, more than 2 have visits lasting longer than 10 minutes.
4. Find the probability that, of 35 randomly chosen people, fewer than 16 have visits lasting less than 8.2 minutes.

5 The weights of letters posted by a certain business are normally distributed with mean 20 g . It is found that the weights of \(94 \%\) of the letters are within 12 g of the mean.

1. Find the standard deviation of the weights of the letters.
2. Find the probability that a randomly chosen letter weighs more than 13 g .
3. Find the probability that at least 2 of a random sample of 7 letters have weights which are more than 12 g above the mean.

5 A triangular spinner has one red side, one blue side and one green side. The red side is weighted so that the spinner is four times more likely to land on the red side than on the blue side. The green side is weighted so that the spinner is three times more likely to land on the green side than on the blue side.

1. Show that the probability that the spinner lands on the blue side is \(\frac { 1 } { 8 }\).
2. The spinner is spun 3 times. Find the probability that it lands on a different coloured side each time.
3. The spinner is spun 136 times. Use a suitable approximation to find the probability that it lands on the blue side fewer than 20 times.

6 There are a large number of students in Luttley College. \(60 \%\) of the students are boys. Students can choose exactly one of Games, Drama or Music on Friday afternoons. It is found that \(75 \%\) of the boys choose Games, \(10 \%\) of the boys choose Drama and the remainder of the boys choose Music. Of the girls, \(30 \%\) choose Games, \(55 \%\) choose Drama and the remainder choose Music.

1. 6 boys are chosen at random. Find the probability that fewer than 3 of them choose Music.
2. 5 Drama students are chosen at random. Find the probability that at least 1 of them is a boy.
3. In a certain country, the daily minimum temperature, in \({ } ^ { \circ } \mathrm { C }\), in winter has the distribution \(\mathrm { N } ( 8,24 )\). Find the probability that a randomly chosen winter day in this country has a minimum temperature between \(7 ^ { \circ } \mathrm { C }\) and \(12 ^ { \circ } \mathrm { C }\). The daily minimum temperature, in \({ } ^ { \circ } \mathrm { C }\), in another country in winter has a normal distribution with mean \(\mu\) and standard deviation \(2 \mu\).
4. Find the proportion of winter days on which the minimum temperature is below zero.
5. 70 winter days are chosen at random. Find how many of these would be expected to have a minimum temperature which is more than three times the mean.
6. The probability of the minimum temperature being above \(6 ^ { \circ } \mathrm { C }\) on any winter day is 0.0735 . Find the value of \(\mu\).

6 Human blood groups are identified by two parts. The first part is \(\mathrm { A } , \mathrm { B } , \mathrm { AB }\) or O and the second part (the Rhesus part) is + or - . In the UK, \(35 \%\) of the population are group \(\mathrm { A } + , 8 \%\) are \(\mathrm { B } + , 3 \%\) are \(\mathrm { AB } +\), \(37 \%\) are \(\mathrm { O } + , 7 \%\) are \(\mathrm { A } - , 2 \%\) are \(\mathrm { B } - , 1 \%\) are \(\mathrm { AB } -\) and \(7 \%\) are \(\mathrm { O } -\).

1. A random sample of 9 people in the UK who are Rhesus + is taken. Find the probability that fewer than 3 are group \(\mathrm { O } +\).
2. A random sample of 150 people in the UK is taken. Find the probability that more than 60 people are group A+.

5 A company set up a display consisting of 20 fireworks. For each firework, the probability that it fails to work is 0.05 , independently of other fireworks.

1. Find the probability that more than 1 firework fails to work. The 20 fireworks cost the company \(\\) 24\( each. 450 people pay the company \)\\( 10\) each to watch the display. If more than 1 firework fails to work they get their money back.
2. Calculate the expected profit for the company.

6 Ana meets her friends once every day. For each day the probability that she is early is 0.05 and the probability that she is late is 0.75 . Otherwise she is on time.

1. Find the probability that she is on time on fewer than 20 of the next 96 days.
2. If she is early there is a probability of 0.7 that she will eat a banana. If she is late she does not eat a banana. If she is on time there is a probability of 0.4 that she will eat a banana. Given that for one particular meeting with friends she does not eat a banana, find the probability that she is on time.

3 A researcher read a magazine article which stated that boys aged 1 to 3 prefer green to orange. It claimed that, when offered a green cube and an orange cube to play with, a boy is more likely to choose the green one. The researcher disagrees with this claim. She believes that boys of this age are equally likely to choose either colour. In order to test her belief, the researcher carried out a hypothesis test at the 5\% significance level. She offered a green cube and an orange cube to each of 10 randomly chosen boys aged 1 to 3 , and recorded the number, \(X\), of boys who chose the green cube. Out of the 10 boys, 8 boys chose the green cube.

1. 1. Assuming that the researcher's belief that either colour cube is equally likely to be chosen is valid, a student correctly calculates that \(\mathrm { P } ( X = 8 ) = 0.0439\), correct to 3 significant figures. He says that, because this value is less than 0.05 , the null hypothesis should be rejected. Explain why this statement is incorrect.
   2. Carry out the test on the researcher's claim that either colour cube is equally likely to be chosen.
2. Another researcher claims that a Type I error was made in carrying out the test. Explain why this cannot be true.
   A similar test, at the \(5 \%\) significance level, was carried out later using 10 other randomly chosen boys aged 1 to 3 .
3. Find the probability of a Type I error.

3 A factory owner models the number of employees who use the factory canteen on any day by the distribution \(\mathrm { B } ( 25 , p )\). In the past the value of \(p\) was 0.8 . A new menu is introduced in the canteen and the owner wants to test whether the value of \(p\) has increased. On a randomly chosen day he notes that the number of employees who use the canteen is 23 .

1. Use the binomial distribution to carry out the test at the \(10 \%\) significance level.
2. Given that there are 30 employees at the factory comment on the suitability of the owner's model. \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-04_2713_33_111_2016} \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-05_2716_29_107_22}

4 In a certain city it is necessary to pass a driving test in order to be allowed to drive a car. The probability of passing the driving test at the first attempt is 0.36 on average. A particular driving instructor claims that the probability of his pupils passing at the first attempt is higher than 0.36 . A random sample of 8 of his pupils showed that 7 passed at the first attempt.

1. Carry out an appropriate hypothesis test to test the driving instructor's claim, using a significance level of \(5 \%\).
2. In fact, most of this random sample happened to be careful and sensible drivers. State which type of error in the hypothesis test (Type I or Type II) could have been made in these circumstances and find the probability of this type of error when a sample of size 8 is used for the test.