5.02d Binomial: mean np and variance np(1-p)

3 On a certain road \(20 \%\) of the vehicles are trucks, \(16 \%\) are buses and the remainder are cars.

1. A random sample of 11 vehicles is taken. Find the probability that fewer than 3 are buses.
2. A random sample of 125 vehicles is now taken. Using a suitable approximation, find the probability that more than 73 are cars.

5 In the holidays Martin spends \(25 \%\) of the day playing computer games. Martin's friend phones him once a day at a randomly chosen time.

1. Find the probability that, in one holiday period of 8 days, there are exactly 2 days on which Martin is playing computer games when his friend phones.
2. Another holiday period lasts for 12 days. State with a reason whether it is appropriate to use a normal approximation to find the probability that there are fewer than 7 days on which Martin is playing computer games when his friend phones.
3. Find the probability that there are at least 13 days of a 40-day holiday period on which Martin is playing computer games when his friend phones.

3 Christa takes her dog for a walk every day. The probability that they go to the park on any day is 0.6 . If they go to the park there is a probability of 0.35 that the dog will bark. If they do not go to the park there is a probability of 0.75 that the dog will bark.

1. Find the probability that they go to the park on more than 5 of the next 7 days.
2. Find the probability that the dog barks on any particular day.
3. Find the variance of the number of times they go to the park in 30 days.

7 The heights that children of a particular age can jump have a normal distribution. On average, 8 children out of 10 can jump a height of more than 127 cm , and 1 child out of 3 can jump a height of more than 135 cm .

1. Find the mean and standard deviation of the heights the children can jump.
2. Find the probability that a randomly chosen child will not be able to jump a height of 145 cm .
3. Find the probability that, of 8 randomly chosen children, at least 2 will be able to jump a height of more than 135 cm .

4 Robert uses his calculator to generate 5 random integers between 1 and 9 inclusive.

1. Find the probability that at least 2 of the 5 integers are less than or equal to 4 . Robert now generates \(n\) random integers between 1 and 9 inclusive. The random variable \(X\) is the number of these \(n\) integers which are less than or equal to a certain integer \(k\) between 1 and 9 inclusive. It is given that the mean of \(X\) is 96 and the variance of \(X\) is 32 .
2. Find the values of \(n\) and \(k\).

7 During the school holidays, each day Khalid either rides on his bicycle with probability 0.6 , or on his skateboard with probability 0.4 . Khalid does not ride on both on the same day. If he rides on his bicycle then the probability that he hurts himself is 0.05 . If he rides on his skateboard the probability that he hurts himself is 0.75 .

1. Find the probability that Khalid hurts himself on any particular day.
2. Given that Khalid hurts himself on a particular day, find the probability that he is riding on his skateboard.
3. There are 45 days of school holidays. Show that the variance of the number of days Khalid rides on his skateboard is the same as the variance of the number of days that Khalid rides on his bicycle.
4. Find the probability that Khalid rides on his skateboard on at least 2 of 10 randomly chosen days in the school holidays.

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 \(n\).

2 The random variable \(X\) has the distribution \(\mathrm { B } ( 400,0.01 )\).

1. Find \(\operatorname { Var } ( 4 X + 2 )\).
2. 1. State an appropriate approximating distribution for \(X\), giving the values of any parameters. Justify your choice of approximating distribution.
   2. Use your approximating distribution to find \(\mathrm { P } ( 2 \leqslant X \leqslant 5 )\).

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 \(n\).

7

1. A large number of spoons and forks made in a factory are inspected. It is found that \(1 \%\) of the spoons and \(1.5 \%\) of the forks are defective. A random sample of 140 items, consisting of 80 spoons and 60 forks, is chosen. Use the Poisson approximation to the binomial distribution to find the probability that the sample contains
   1. at least 1 defective spoon and at least 1 defective fork,
   2. fewer than 3 defective items.
2. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). It is given that $$\mathrm { P } ( X = 1 ) = p \quad \text { and } \quad \mathrm { P } ( X = 2 ) = 1.5 p$$ where \(p\) is a non-zero constant. Find the value of \(\lambda\) and hence find the value of \(p\).

1 On average, 1 clover plant in 10000 has four leaves instead of three.

1. Use an approximating distribution to calculate the probability that, in a random sample of 2000 clover plants, more than 2 will have four leaves.
2. Justify your approximating distribution.

7 The independent variables \(X\) and \(Y\) are such that \(X \sim \mathrm {~B} ( 10,0.8 )\) and \(Y \sim \mathrm { Po } ( 3 )\). Find

1. \(\mathrm { E } ( 7 X + 5 Y - 2 )\),
2. \(\operatorname { Var } ( 4 X - 3 Y + 3 )\),
3. \(\mathrm { P } ( 2 X - Y = 18 )\).

6 The discrete random variable \(X\) takes the values 0 and 1 with \(\mathrm { P } ( X = 0 ) = q\) and \(\mathrm { P } ( X = 1 ) = p\), where \(p + q = 1\).

1. Write down the probability generating function of \(X\). The sum of \(n\) independent observations of \(X\) is denoted by \(S\).
2. Write down the probability generating function of \(S\), and name the distribution of \(S\).
3. Use the probability generating function of \(S\) to find \(\mathrm { E } ( S )\) and \(\operatorname { Var } ( S )\).
4. The independent random variables \(Y\) and \(Z\) are such that \(Y\) has the distribution \(\mathrm { B } \left( 10 , \frac { 1 } { 2 } \right)\), and \(Z\) has probability generating function \(\mathrm { e } ^ { - ( 1 - t ) }\). Find the probability that the sum of one random observation of \(Y\) and one random observation of \(Z\) is equal to 2 .

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

1. Show, from the definition, that the probability generating function of \(X\) is \(( q + p t ) ^ { n }\), where \(q = 1 - p\).
2. The independent random variable \(Y\) has the distribution \(\mathrm { B } ( 2 n , p )\) and \(T = X + Y\). Use probability generating functions to determine the distribution of \(T\), giving its parameters.

1 Independent random variables \(X\) and \(Y\) have distributions \(\mathrm { B } ( 7 , p )\) and \(\mathrm { B } ( 8 , p )\) respectively.

1. Explain why \(X + Y \sim \mathrm {~B} ( 15 , p )\).
2. Find \(\mathrm { P } ( X = 2 \mid X + Y = 5 )\).

7 Each question on a multiple-choice examination paper has \(n\) possible responses, only one of which is correct. Joni takes the paper and has probability \(p\), where \(0 < p < 1\), of knowing the correct response to any question, independently of any other. If she knows the correct response she will choose it, otherwise she will choose randomly from the \(n\) possibilities. The events \(K\) and \(A\) are 'Joni knows the correct response' and 'Joni answers correctly' respectively.

1. Show that \(\mathrm { P } ( A ) = \frac { q + n p } { n }\), where \(q = 1 - p\).
2. Find \(P ( K \mid A )\). A paper with 100 questions has \(n = 4\) and \(p = 0.5\). Each correct response scores 1 and each incorrect response scores - 1 .
3. (a) Joni answers all the questions on the paper and scores 40 . How many questions did she answer correctly?
   (b) By finding the distribution of the number of correct answers, or otherwise, find the probability that Joni scores at least 40 on the paper using her strategy.

6 In each of 38 randomly selected weeks of the English Premier Football League there were 10 matches. Table 1 summarises the number of home wins in 10 matches, \(X\), and the corresponding number of weeks. \begin{table}[h] \end{table} A researcher investigates whether \(X\) can be modelled by the distribution \(\mathrm { B } ( 10 , p )\). He calculates the expected frequencies using a value of \(p\) obtained from the sample mean.

1. Show that \(p = 0.45\). Table 2 shows the observed and expected number of weeks. \begin{table}[h]

   Number of home wins	0	1	2	3	4	5	6	7	8	9	10	Totals
   Observed number of weeks	0	1	2	8	8	9	7	1	2	0	0	38
   Expected number of weeks	0.096	0.788	2.899	6.326	9.058	8.893	6.064	2.835	0.870	0.158	0.013	38

   \captionsetup{labelformat=empty} \caption{Table 2
2. Show how the value of 2.835 for 7 home wins is obtained.}