5.02b Expectation and variance: discrete random variables

2 The number of phone calls, \(X\), received per day by Sarah has the following probability distribution.

1. Find the value of \(k\).
2. Find the mode of \(X\).
3. Find the probability that the number of phone calls received by Sarah on any particular day is more than the mean number of phone calls received per day.

6 Nadia is very forgetful. Every time she logs in to her online bank she only has a \(40 \%\) chance of remembering her password correctly. She is allowed 3 unsuccessful attempts on any one day and then the bank will not let her try again until the next day.

1. Draw a fully labelled tree diagram to illustrate this situation.
2. Let \(X\) be the number of unsuccessful attempts Nadia makes on any day that she tries to log in to her bank. Copy and complete the following table to show the probability distribution of \(X\).

   \(x\)	0	1	2	3
   \(\mathrm { P } ( X = x )\)		0.24

3. Calculate the expected number of unsuccessful attempts made by Nadia on any day that she tries to \(\log\) in.

6 A fair spinner \(A\) has edges numbered \(1,2,3,3\). A fair spinner \(B\) has edges numbered \(- 3 , - 2 , - 1,1\). Each spinner is spun. The number on the edge that the spinner comes to rest on is noted. Let \(X\) be the sum of the numbers for the two spinners.

1. Copy and complete the table showing the possible values of \(X\).

   	Spinner \(A\)
   \cline { 2 - 6 }		1	2	3	3
   Spinner \(B\)	- 2
   - 2			1
   - 1
   1

2. Draw up a table showing the probability distribution of \(X\).
3. Find \(\operatorname { Var } ( X )\).
4. Find the probability that \(X\) is even, given that \(X\) is positive.

2 Two fair six-sided dice with faces numbered 1, 2, 3, 4, 5, 6 are thrown and the two scores are noted. The difference between the two scores is defined as follows.

• If the scores are equal the difference is zero.
• If the scores are not equal the difference is the larger score minus the smaller score.

Find the expectation of the difference between the two scores.

3 Visitors to a Wildlife Park in Africa have independent probabilities of 0.9 of seeing giraffes, 0.95 of seeing elephants, 0.85 of seeing zebras and 0.1 of seeing lions.

1. Find the probability that a visitor to the Wildlife Park sees all these animals.
2. Find the probability that, out of 12 randomly chosen visitors, fewer than 3 see lions.
3. 50 people independently visit the Wildlife Park. Find the mean and variance of the number of these people who see zebras.

2 Noor has 3 T-shirts, 4 blouses and 5 jumpers. She chooses 3 items at random. The random variable \(X\) is the number of T-shirts chosen.

1. Show that the probability that Noor chooses exactly one T-shirt is \(\frac { 27 } { 55 }\).
2. Draw up the probability distribution table for \(X\).

6 The weights of bananas in a fruit shop have a normal distribution with mean 150 grams and standard deviation 50 grams. Three sizes of banana are sold. Small: under 95 grams
Medium: between 95 grams and 205 grams
Large: over 205 grams

1. Find the proportion of bananas that are small.
2. Find the weight exceeded by \(10 \%\) of bananas. The prices of bananas are 10 cents for a small banana, 20 cents for a medium banana and 25 cents for a large banana.
3. (a) Show that the probability that a randomly chosen banana costs 20 cents is 0.7286 .
   (b) Calculate the expected total cost of 100 randomly chosen bananas.

1 The discrete random variable \(X\) has the following probability distribution. Given that \(\mathrm { E } ( X ) = 3.05\), find the values of \(p\) and \(q\).

4 A fair die with faces numbered \(1,2,2,2,3,6\) is thrown. The score, \(X\), is found by squaring the number on the face the die shows and then subtracting 4.

1. Draw up a table to show the probability distribution of \(X\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

2 A random variable \(X\) has the probability distribution shown in the following table, where \(p\) is a constant.

1. Find the value of \(p\).
2. Given that \(\mathrm { E } ( X ) = 1.15\), find \(\operatorname { Var } ( X )\).

6 A fair red spinner has 4 sides, numbered 1,2,3,4. A fair blue spinner has 3 sides, numbered 1,2,3. When a spinner is spun, the score is the number on the side on which it lands. The spinners are spun at the same time. The random variable \(X\) denotes the score on the red spinner minus the score on the blue spinner.

1. Draw up the probability distribution table for \(X\).
2. Find \(\operatorname { Var } ( X )\).
3. Find the probability that \(X\) is equal to 1 , given that \(X\) is non-zero.

2 A fair 6 -sided die has the numbers \(- 1 , - 1,0,0,1,2\) on its faces. A fair 3 -sided spinner has edges numbered \(- 1,0,1\). The die is thrown and the spinner is spun. The number on the uppermost face of the die and the number on the edge on which the spinner comes to rest are noted. The sum of these two numbers is denoted by \(X\).

1. Draw up a table showing the probability distribution of \(X\).
2. Find \(\operatorname { Var } ( X )\).

4 In a probability distribution the random variable \(X\) takes the values \(- 1,0,1,2,4\). The probability distribution table for \(X\) is as follows.

1. Find the value of \(p\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Given that \(X\) is greater than zero, find the probability that \(X\) is equal to 2 .

5 A fair red spinner has four sides, numbered 1, 2, 3, 3. A fair blue spinner has three sides, numbered \(- 1,0,2\). When a spinner is spun, the score is the number on the side on which it lands. The spinners are spun at the same time. The random variable \(X\) denotes the score on the red spinner minus the score on the blue spinner.

1. Draw up the probability distribution table for \(X\).
2. Find \(\operatorname { Var } ( X )\).

6 A box contains 3 red balls and 5 white balls. One ball is chosen at random from the box and is not returned to the box. A second ball is now chosen at random from the box.

1. Find the probability that both balls chosen are red.
2. Show that the probability that the balls chosen are of different colours is \(\frac { 15 } { 28 }\).
3. Given that the second ball chosen is red, find the probability that the first ball chosen is red.
   The random variable \(X\) denotes the number of red balls chosen.
4. Draw up the probability distribution table for \(X\).
5. Find \(\operatorname { Var } ( X )\).

6 Nadia is very forgetful. Every time she logs in to her online bank she only has a \(40 \%\) chance of remembering her password correctly. She is allowed 3 unsuccessful attempts on any one day and then the bank will not let her try again until the next day.

1. Draw a fully labelled tree diagram to illustrate this situation.
2. Let \(X\) be the number of unsuccessful attempts Nadia makes on any day that she tries to log in to her bank. Complete the following table to show the probability distribution of \(X\).

   \(x\)	0	1	2	3
   \(\mathrm { P } ( X = x )\)		0.24

3. Calculate the expected number of unsuccessful attempts made by Nadia on any day that she tries to \(\log\) in.

4 A fair spinner has five sides numbered \(1,2,3,4,5\). The score on one spin is denoted by \(X\).

1. Show that \(\operatorname { Var } ( X ) = 2\).
   Fiona has another spinner, also with five sides numbered \(1,2,3,4,5\). She suspects that it is biased so that the expected score is less than 3 . In order to test her suspicion, she plans to spin her spinner 40 times. If the mean score is less than 2.6 she will conclude that her spinner is biased in this way.
2. Find the probability of a Type I error.
3. State what is meant by a Type II error in this context.

3 In the data-entry department of a certain firm, it is known that \(0.12 \%\) of data items are entered incorrectly, and that these errors occur randomly and independently.

1. A random sample of 3600 data items is chosen. The number of these data items that are incorrectly entered is denoted by \(X\).
   1. State the distribution of \(X\), including the values of any parameters.
   2. State an appropriate approximating distribution for \(X\), including the values of any parameters. Justify your choice of approximating distribution.
   3. Use your approximating distribution to find \(\mathrm { P } ( X > 2 )\).
2. Another large random sample of \(n\) data items is chosen. The probability that the sample contains no data items that are entered incorrectly is more than 0.1 . Use an approximating distribution to find the largest possible value of \(n\).

4 The score on one spin of a 5 -sided spinner is denoted by the random variable \(X\) with probability distribution as shown in the table.

1. Show that \(\operatorname { Var } ( X ) = 1.2\).
   The spinner is spun 200 times. The score on each spin is noted and the mean, \(\bar { X }\), of the 200 scores is found.
2. Given that \(\mathrm { P } ( \bar { X } > a ) = 0.1\), find the value of \(a\).
3. Explain whether it was necessary to use the Central Limit theorem in your answer to part (b).
4. Johann has another, similar, spinner. He suspects that it is biased so that the mean score is less than 2 . He spins his spinner 200 times and finds that the mean of the 200 scores is 1.86 . Given that the variance of the score on one spin of this spinner is also 1.2 , test Johann's suspicion at the 5\% significance level.

1 A fair coin is tossed 5 times and the number of heads is recorded.

1. The random variable \(X\) is the number of heads. State the mean and variance of \(X\).
2. The number of heads is doubled and denoted by the random variable \(Y\). State the mean and variance of \(Y\).

5 Most plants of a certain type have three leaves. However, it is known that, on average, 1 in 10000 of these plants have four leaves, and plants with four leaves are called 'lucky'. The number of lucky plants in a random sample of 25000 plants is denoted by \(X\).

1. State, with a justification, an approximating distribution for \(X\), giving the values of any parameters.
   Use your approximating distribution to answer parts (b) and (c).
2. Find \(\mathrm { P } ( X \leqslant 3 )\).
3. Given that \(\mathrm { P } ( X = k ) = 2 \mathrm { P } ( X = k + 1 )\), find \(k\).
   The number of lucky plants in a random sample of \(n\) plants, where \(n\) is large, is denoted by \(Y\).
4. Given that \(\mathrm { P } ( Y \geqslant 1 ) = 0.963\), correct to 3 significant figures, use a suitable approximating distribution to find the value of \(n\).

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.

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.

4 On average, 1 in 400 microchips made at a certain factory are faulty. The number of faulty microchips in a random sample of 1000 is denoted by \(X\).

1. State the distribution of \(X\), giving the values of any parameters.
2. State an approximating distribution for \(X\), giving the values of any parameters.
3. Use this approximating distribution to find each of the following.
   1. \(\mathrm { P } ( X = 4 )\).
   2. \(\mathrm { P } ( 2 \leqslant X \leqslant 4 )\).
4. Use a suitable approximating distribution to find the probability that, in a random sample of 700 microchips, there will be at least 1 faulty one.

2 In a probability distribution the random variable \(X\) takes the value \(x\) with probability \(k x\), where \(x\) takes values \(1,2,3,4,5\) only.

1. Draw up a probability distribution table for \(X\), in terms of \(k\), and find the value of \(k\).
2. Find \(\mathrm { E } ( X )\).