2.04c Calculate binomial probabilities

13 At a certain factory Christmas tree decorations are packed in boxes of 10 . The quality control manager collects a random sample of 100 boxes of decorations and records the number of decorations in each box which are damaged. His results are displayed in Fig. 13.1. \begin{table}[h] \end{table}

1. Calculate
   It is believed that the number of damaged decorations in a box of 10, \(X\), may be modelled by a binomial distribution such that \(\mathrm { X } \sim \mathrm { B } ( \mathrm { n } , \mathrm { p } )\).
2. State suitable values for \(n\) and \(p\).
3. Use the binomial model to complete the copy of Fig. 13.2 in the Printed Answer Booklet, giving your answers correct to \(\mathbf { 1 }\) decimal place. \begin{table}[h]

   Number of damaged decorations	0	1	2	3	4	5 or more
   Observed number of boxes	19	35	28	13	5	0
   Expected number of boxes

   \captionsetup{labelformat=empty} \caption{Fig. 13.2}
   \end{table}
4. Explain whether the model is a good fit for these data.

The weights of packages delivered to Susie are normally distributed with a mean of 510 grams and a standard deviation of 45 grams.

1. Find the probability that a randomly selected package delivered to Susie weighs less than 450 grams. The heaviest 5\% of packages delivered to Susie are delivered by Rav in his van, the others are delivered by Taruni on foot.
2. Find the weight of the lightest package that Rav would deliver to Susie. Susie randomly selects a package from those delivered by Taruni.
3. Find the probability that this package weighs more than 450 grams. On Tuesday there are 5 packages delivered to Susie.
4. Find the probability that 4 are delivered by Taruni and 1 is delivered by Rav.

1. Gill buys a bag of logs to use in her stove. The lengths, \(l \mathrm {~cm}\), of the 88 logs in the bag are summarised in the table below.

A histogram is drawn to represent these data.
The bar representing logs with length \(27 < l \leqslant 30\) has a width of 1.5 cm and a height of 4 cm .

1. Calculate the width and height of the bar representing log lengths of \(20 < l \leqslant 25\)
2. Use linear interpolation to estimate the median of \(l\) The maximum length of log Gill can use in her stove is 26 cm .
   Gill estimates, using linear interpolation, that \(x\) logs from the bag will fit into her stove.
3. Show that \(x = 62\) Gill randomly selects 4 logs from the bag.
4. Using \(x = 62\), find the probability that all 4 logs will fit into her stove. The weights, \(W\) grams, of the logs in the bag are coded using \(y = 0.5 w - 255\) and summarised by $$n = 88 \quad \sum y = 924 \quad \sum y ^ { 2 } = 12862$$
5. Calculate
   1. the mean of \(W\)
   2. the variance of \(W\)

7. A packing plant fills bags with cement. The weight \(X \mathrm {~kg}\) of a bag of cement can be modelled by a normal distribution with mean 50 kg and standard deviation 2 kg .

1. Find \(\mathrm { P } ( X > 53 )\).
2. Find the weight that is exceeded by \(99 \%\) of the bags. Three bags are selected at random.
3. Find the probability that two weigh more than 53 kg and one weighs less than 53 kg .

7. Farmer Adam grows potatoes. The weights of potatoes, in grams, grown by Adam are normally distributed with a mean of 140 g and a standard deviation of 40 g . Adam cannot sell potatoes with a weight of less than 92 g .

1. Find the percentage of potatoes that Adam grows but cannot sell. The upper quartile of the weight of potatoes sold by Adam is \(q _ { 3 }\)
2. Find the probability that the weight of a randomly selected potato grown by Adam is more than \(q _ { 3 }\)
3. Find the lower quartile, \(q _ { 1 }\), of the weight of potatoes sold by Adam. Betty selects a random sample of 3 potatoes sold by Adam.
4. Find the probability that one weighs less than \(q _ { 1 }\), one weighs more than \(q _ { 3 }\) and one has a weight between \(q _ { 1 }\) and \(q _ { 3 }\)

   END

4. A customer wishes to withdraw money from a cash machine. To do this it is necessary to type a PIN number into the machine. The customer is unsure of this number. If the wrong number is typed in, the customer can try again up to a maximum of four attempts in total. Attempts to type in the correct number are independent and the probability of success at each attempt is 0.6 .

1. Show that the probability that the customer types in the correct number at the third attempt is 0.096 .
   (2 marks)
   The random variable \(A\) represents the number of attempts made to type in the correct PIN number, regardless of whether or not the attempt is successful.
2. Find the probability distribution of \(A\).
3. Calculate the probability that the customer types in the correct number in four or fewer attempts.
4. Calculate \(\mathrm { E } ( A )\) and \(\operatorname { Var } ( A )\).
5. Find \(\mathrm { F } ( 1 + \mathrm { E } ( A ) )\).

1. The probability of a leaf cutting successfully taking root is 0.05

Find the probability that, in a batch of 10 randomly selected leaf cuttings, the number taking root will be

1. 1. exactly 1
   2. more than 2 A second random sample of 160 leaf cuttings is selected.
2. Using a suitable approximation, estimate the probability of at least 10 leaf cuttings taking root.
   

5. A school photocopier breaks down randomly at a rate of 15 times per year.

1. Find the probability that there will be exactly 3 breakdowns in the next month.
2. Show that the probability that there will be at least 2 breakdowns in the next month is 0.355 to 3 decimal places.
3. Find the probability of at least 2 breakdowns in each of the next 4 months. The teachers would like a new photocopier. The head teacher agrees to monitor the situation for the next 12 months. The head teacher decides he will buy a new photocopier if there is more than 1 month when the photocopier has at least 2 breakdowns.
4. Find the probability that the head teacher will buy a new photocopier.

6. The Headteacher of a school claims that \(30 \%\) of parents do not support a new curriculum. In a survey of 20 randomly selected parents, the number, \(X\), who do not support the new curriculum is recorded. Assuming that the Headteacher's claim is correct, find

1. the probability that \(X = 5\)
2. the mean and the standard deviation of \(X\) The Director of Studies believes that the proportion of parents who do not support the new curriculum is greater than \(30 \%\). Given that in the survey of 20 parents 8 do not support the new curriculum,
3. test, at the \(5 \%\) level of significance, the Director of Studies' belief. State your hypotheses clearly. The teachers believe that the sample in the original survey was biased and claim that only \(25 \%\) of the parents are in support of the new curriculum. A second random sample, of size \(2 n\), is taken and exactly half of this sample supports the new curriculum. A test is carried out at a 10\% level of significance of the teachers' belief using this sample of size \(2 n\) Using the hypotheses \(\mathrm { H } _ { 0 } : p = 0.25\) and \(\mathrm { H } _ { 1 } : p > 0.25\)
4. find the minimum value of \(n\) for which the outcome of the test is that the teachers' belief is rejected.

7. A multiple choice examination paper has \(n\) questions where \(n > 30\) Each question has 5 answers of which only 1 is correct. A pass on the paper is obtained by answering 30 or more questions correctly. The probability of obtaining a pass by randomly guessing the answer to each question should not exceed 0.0228 Use a normal approximation to work out the greatest number of questions that could be used.

The continuous random variable \(W\) has the normal distribution \(\mathrm { N } \left( 32,4 { } ^ { 2 } \right)\)

1. Write down the value of \(\mathrm { P } ( W = 36 )\) The discrete random variable \(X\) has the binomial distribution \(\mathrm { B } ( 20,0.45 )\)
2. Find \(\mathrm { P } ( X = 8 )\)
3. Find the probability that \(X\) lies within one standard deviation of its mean.
   

1. In the manufacture of cloth in a factory, defects occur randomly in the production process at a rate of 2 per \(5 \mathrm {~m} ^ { 2 }\)

The quality control manager randomly selects 12 pieces of cloth each of area \(15 \mathrm {~m} ^ { 2 }\).

1. Find the probability that exactly half of these 12 pieces of cloth will contain at most 7 defects. The factory introduces a new procedure to manufacture the cloth. After the introduction of this new procedure, the manager takes a random sample of \(25 \mathrm {~m} ^ { 2 }\) of cloth from the next batch produced to test if there has been any change in the rate of defects.
2. 1. Write down suitable hypotheses for this test.
   2. Describe a suitable test statistic that the manager should use.
   3. Explain what is meant by the critical region for this test.
3. Using a 5\% level of significance, find the critical region for this test. You should choose the largest critical region for which the probability in each tail is less than 2.5\%
4. Find the actual significance level for this test.

1. Albert uses scales in his kitchen to weigh some fruit.

The random variable \(D\) represents, in grams, the weight of the fruit given by the scales minus the true weight of the fruit. The random variable \(D\) is uniformly distributed over the interval \([ - 2.5,2.5 ]\)

1. Specify the probability density function of \(D\)
2. Find the standard deviation of \(D\) Albert weighs a banana on the scales.
3. Write down the probability that the weight given by the scales equals the true weight of the banana.
4. Find the probability that the weight given by the scales is within 1 gram of the banana's true weight. Albert weighs 10 bananas on the scales, one at a time.
5. Find the probability that the weight given by the scales is within 1 gram of the true weight for at least 6 of the bananas.

5. A delivery company loses packages randomly at a mean rate of 10 per month. The probability that the delivery company loses more than 12 packages in a randomly selected month is \(p\)

1. Find the value of \(p\) The probability that the delivery company loses more than \(k\) packages in a randomly selected month is at least \(2 p\)
2. Find the largest possible value of \(k\) In a randomly selected month,
3. find the probability that exactly 4 packages were lost in each half of the month. In a randomly selected two-month period, 21 packages were lost.
4. Find the probability that at least 10 packages were lost in each of these two months.
5. Using a suitable approximation, find the probability that more than 27 packages are lost during a randomly selected 4-month period.

1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by

$$f ( x ) = \begin{cases} \frac { 1 } { 16 } x ^ { 2 } & 1 \leqslant x < 3 \\ k ( 4 - x ) & 3 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 11 } { 12 }\)
2. Sketch \(\mathrm { f } ( x )\) for \(1 \leqslant x \leqslant 4\)
3. Write down the mode of \(X\) Given that \(\mathrm { E } ( X ) = \frac { 25 } { 9 }\)
4. use algebraic integration to find \(\operatorname { Var } ( X )\), giving your answer to 3 significant figures. The cumulative distribution function of \(X\) is given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1 \\ \frac { 1 } { 48 } \left( x ^ { 3 } + c \right) & 1 \leqslant x < 3 \\ \frac { 11 } { 12 } \left( 4 x - \frac { 1 } { 2 } x ^ { 2 } + d \right) & 3 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$
5. 1. Find the exact value of \(C\)
   2. Find the exact value of \(d\)
6. Calculate, to 3 significant figures, the upper quartile of \(X\)
   \includegraphics[max width=\textwidth, alt={}]{a814156d-6945-4601-9cae-d28d8ae0db1e-28_2632_1826_121_121}

1. During morning hours, employees arrive randomly at an office drinks dispenser at a rate of 2 every 10 minutes.

The number of employees arriving at the drinks dispenser is assumed to follow a Poisson distribution.

1. Find the probability that fewer than 5 employees arrive at the drinks dispenser during a 10-minute period one morning. During a 30 -minute period one morning, the probability that \(n\) employees arrive at the drinks dispenser is the same as the probability that \(n + 1\) employees arrive at the drinks dispenser.
2. Find the value of \(n\) During a 45-minute period one morning, the probability that between \(c\) and 12, inclusive, employees arrive at the drinks dispenser is 0.8546
3. Find the value of \(C\)
4. Find the probability that exactly 2 employees arrive at the drinks dispenser in exactly 4 of the 6 non-overlapping 10-minute intervals between 10 am and 11am one morning.

1. At a shop, past figures show that \(35 \%\) of customers pay by credit card. Following the shop's decision to no longer charge a fee for using a credit card, a random sample of 20 customers is taken and 11 are found to have paid by credit card.

Hadi believes that the proportion of customers paying by credit card is now greater than 35\%

1. Test Hadi's belief at the \(5 \%\) level of significance. State your hypotheses clearly. For a random sample of 20 customers,
2. show that 11 lies less than 2 standard deviations above the mean number of customers paying by credit card.
   You may assume that \(35 \%\) is the true proportion of customers who pay by credit card.
   

1. Jim farms oysters in a particular lake. He knows from past experience that \(5 \%\) of young oysters do not survive to be harvested.

In a random sample of 30 young oysters, the random variable \(X\) represents the number that do not survive to be harvested.

1. Write down a suitable model for the distribution of \(X\).
2. State an assumption that has been made for the model in part (a).
3. Find the probability that
   1. exactly 24 young oysters do survive to be harvested,
   2. at least 3 young oysters do not survive to be harvested. A second random sample, of 200 young oysters, is taken. The probability that at least \(n\) of these young oysters do not survive to be harvested is more than 0.8
4. Using a suitable approximation, find the maximum value of \(n\). Jim believes that the level of salt in the lake water has changed and it has altered the survival rate of his oysters. He takes a random sample of 25 young oysters and places them in the lake.
   When Jim harvests the oysters, he finds that 21 do survive to be harvested.
5. Use a suitable test, at the \(5 \%\) level of significance, to assess whether or not there is evidence that the proportion of oysters not surviving to be harvested is more than \(5 \%\). State your hypotheses clearly.

2. The distance, in metres, a novice tightrope artist, walking on a wire, walks before falling is modelled by the random variable \(W\) with cumulative distribution function $$\mathrm { F } ( w ) = \left\{ \begin{array} { c c } 0 & w < 0 \\ \frac { 1 } { 3 } \left( w - \frac { w ^ { 4 } } { 256 } \right) & 0 \leqslant w \leqslant 4 \\ 1 & w > 4 \end{array} \right.$$

1. Find the probability that a novice tightrope artist, walking on the wire, walks at least 3.5 metres before falling. A random sample of 30 novice tightrope artists is taken.
2. Find the probability that more than 1 of these novice tightrope artists, walking on the wire, walks at least 3.5 metres before falling. Given \(\mathrm { E } ( W ) = 1.6\)
3. use algebraic integration to find \(\operatorname { Var } ( W )\) DO NOT WRITEIN THIS AREA

3. The number of water fleas, in 100 ml of pond water, has a Poisson distribution with mean 7

1. Find the probability that a sample of 100 ml of the pond water does not contain exactly 4 water fleas. Aja collects 5 separate samples, each of 100 ml , of the pond water.
2. Find the probability that exactly 1 of these samples contains exactly 4 water fleas. Using a normal approximation, the probability that more than 3 water fleas will be found in a random sample of \(n \mathrm { ml }\) of the pond water is 0.9394 correct to 4 significant figures.
3. 1. Show that \(n - 1.55 \sqrt { \frac { n } { 0.07 } } - 50 = 0\)
   2. Hence find the value of \(n\) After the pond has been cleaned, the number of water fleas in a 100 ml random sample of the pond water is 15
4. Using a suitable test, at the \(1 \%\) level of significance, assess whether or not there is evidence that the number of water fleas per 100 ml of the pond water has increased. State your hypotheses clearly. \includegraphics[max width=\textwidth, alt={}, center]{f63c39df-cfc9-4a6b-838d-67613710b0ce-11_2255_50_314_34}
   

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

1 A local pottery makes cups. The number of faulty cups made by the pottery in a week follows a Poisson distribution with a mean of 6 In a randomly chosen week, the probability that there will be at least \(x\) faulty cups made is 0.1528

1. Find the value of \(x\)
2. Use a normal approximation to find the probability that in 6 randomly chosen weeks the total number of faulty cups made is fewer than 32 A week is called a "poor week" if at least \(x\) faulty cups are made, where \(x\) is the value found in part (a).
3. Find the probability that in 50 randomly chosen weeks, more than 1 is a "poor week".

5 Applicants for a pilot training programme with a passenger airline are screened for colour blindness. Past records show that the proportion of applicants identified as colour blind is 0.045

1. Write down a suitable model for the distribution of the number of applicants identified as colour blind from a total of \(n\) applicants.
2. State one assumption necessary for this distribution to be a suitable model of this situation.
3. Using a suitable approximation, find the probability that exactly 5 out of 120 applicants are identified as colour blind.
4. Explain why the approximation that you used in part (c) is appropriate. Jaymini claims that 75\% of all applicants for this training programme go on to become pilots. From a random sample of 96 applicants for this training programme 67 go on to become pilots.
5. Using a suitable approximation, test Jaymini's claim at the \(5 \%\) level of significance. State your hypotheses clearly.

1. Superbounce is a manufacturer of tennis balls.

It knows from past records that 10\% of its tennis balls fail a bounce test.

1. Find the probability that from a random sample of 10 of these tennis balls
   1. at least 4 fail the bounce test
   2. more than 1 but fewer than 5 fail the bounce test. The managing director makes changes to the production process and claims that these changes will reduce the probability of its tennis balls failing the bounce test. After the changes were made a random sample of 50 of the tennis balls were tested and it was found that 2 failed the bounce test.
2. Test, at the \(5 \%\) significance level, whether or not this result supports the managing director's claim. In a second random sample of \(n\) tennis balls it was found that none failed the bounce test. As a result of this sample, the managing director's claim is supported at the 1\% significance level.
3. Find the smallest possible value of \(n\)

4 A bag contains 50 counters, each with one of the numbers 4,7 or 10 written on it in the ratio \(2 : 3 : 5\) respectively. A random sample of 2 counters is taken from the bag. The numbers on the 2 counters are recorded as \(D _ { 1 }\) and \(D _ { 2 }\) The random variable \(M\) represents the mean of \(D _ { 1 }\) and \(D _ { 2 }\)

1. Show that \(\mathrm { P } ( M = 4 ) = \frac { 9 } { 245 }\)
2. Find the sampling distribution of \(M\) A random sample of \(n\) sets of 2 counters is taken. The random variable \(T\) represents the number of these \(n\) sets of 2 counters that have a mean of 4 Given that each set of 2 counters is replaced after it is drawn,
3. calculate the minimum value of \(n\) such that \(\mathrm { P } ( T = 0 ) < 0.15\)

5 A receptionist receives incoming telephone calls and should connect them to the appropriate department. The probability of them being connected to the wrong department on the first attempt is 0.05 A random sample of 8 calls is taken.

1. Find the probability that at least 2 of these calls are connected to the wrong department on the first attempt. The receptionist receives 1000 calls each day.
2. Use a Poisson approximation to find the probability that exactly 45 callers are connected to the wrong department on the first attempt in a day. The total time, \(T\) seconds, taken for a call to be answered by a department has a continuous uniform distribution over the interval [10,50]
3. Find \(\mathrm { P } ( T > 16 )\) The number of calls the receptionist receives in a one-minute interval is modelled by a Poisson distribution with mean 6 The receptionist receives a call from Jia and tries to connect it to the right department.
4. Find the probability that in the next 40 seconds Jia's call is answered by the right department on the first attempt and the receptionist has received no other calls.