Multiple independent binomial calculations

7 On any given day, the probability that Moena messages her friend Pasha is 0.72 .

1. Find the probability that for a random sample of 12 days Moena messages Pasha on no more than 9 days.
2. Moena messages Pasha on 1 January. Find the probability that the next day on which she messages Pasha is 5 January.
3. Use an approximation to find the probability that in any period of 100 days Moena messages Pasha on fewer than 64 days.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 Sam is a member of a soccer club. She is practising scoring goals. The probability that Sam will score a goal on any attempt is 0.7 , independently of all other attempts.

1. Sam makes 10 attempts at scoring goals. Find the probability that Sam will score goals on fewer than 8 of these attempts.
2. Find the probability that Sam's first successful attempt will be before her 5th attempt.
3. Wei is a member of the same soccer club. He is also practising scoring goals. The probability that Wei will score a goal on any attempt is 0.6 , independently of all other attempts. Wei is going to keep making attempts until he scores 3 goals.
   Find the probability that he scores his third goal on his 7th attempt.

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.

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.

7 Each day Annabel eats rice, potato or pasta. Independently of each other, the probability that she eats rice is 0.75 , the probability that she eats potato is 0.15 and the probability that she eats pasta is 0.1 .

1. Find the probability that, in any week of 7 days, Annabel eats pasta on exactly 2 days.
2. Find the probability that, in a period of 5 days, Annabel eats rice on 2 days, potato on 1 day and pasta on 2 days.
3. Find the probability that Annabel eats potato on more than 44 days in a year of 365 days.

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. A random sample of 6 people is chosen from the audience.
     (a) Find the probability that no more than 2 of the 6 people are Notes supporters.
     (b) Find the probability that none of the 6 people support either of these choirs.
  2. A random sample of 240 people is chosen from the audience. Use a suitable approximation to find the probability that fewer than 50 do not support either of the choirs.
     

If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 The probability that an online order from a supermarket chain has at least one item missing when delivered is 0.06 . Online orders are 'incomplete' if they contain substitute items and/or have at least one item missing when delivered. The probability that an order is incomplete is 0.15 .

1. Calculate the probability that exactly 2 out of a random sample of 26 online orders have at least one item missing when delivered.
2. Determine the probability that the number of incomplete orders in a random sample of 50 online orders is:
   1. fewer than 10 ;
   2. more than 5;
   3. more than 6 but fewer than 12 .
3. Farokh, the manager of one of the supermarket's stores, examines 50 randomly selected online orders from each of a random sample of 100 of the store's customers. He records, for each of the 50 orders, the number, \(x\), that were incomplete. His summarised results, correct to three significant figures, for the 100 customers selected are $$\bar { x } = 4.33 \text { and } s ^ { 2 } = 3.94$$ Use this information to compare the performance of the store managed by Farokh with that of the supermarket chain as a whole.
   [0pt] [5 marks]

3 At a pottery which manufactures mugs, it is known that \(5 \%\) of mugs are faulty. The mugs are produced in batches of 20 . Faults are modelled as occurring randomly and independently. The number of faulty mugs in a batch is denoted by the random variable \(X\).

1. Determine \(\mathrm { P } ( X \geqslant 2 )\).
2. Find \(\operatorname { Var } ( X )\). Independently of the mugs, the pottery also manufactures cups, and it is known that \(7 \%\) of cups are faulty. The cups are produced in batches of 30 . Faults are modelled as occurring randomly and independently. The number of faulty cups in a batch is denoted by the random variable \(Y\).
3. Determine the standard deviation of \(X + Y\). When 10 batches of cups have been produced, a sample of 15 cups is tested to ensure that the handles of the cups are properly attached.
4. Explain why it might not be sensible to select a sample of 15 cups from the same batch.

4. A company always sends letters by second class post unless they are marked first class. Over a long period of time it has been established that \(20 \%\) of letters to be posted are marked first class.
In a random selection of 10 letters to be posted, find the probability that the number marked first class is

1. at least 3,
2. fewer than 2. One Monday morning there are only 12 first class stamps. Given that there are 70 letters to be posted that day,
3. use a suitable approximation to find the probability that there are enough first class stamps.
4. State an assumption about these 70 letters that is required in order to make the calculation in part (c) valid.

6 Customers at a supermarket can pay at a checkout either by cash, debit card or credit card.

1. The probability that a customer pays by cash is 0.22 . Calculate the probability that exactly 2 customers from a random sample of 24 customers pay by cash.
2. The probability that a customer pays by debit card is 0.45 . Determine the probability that the number of customers who pay by debit card from a random sample of \(\mathbf { 4 0 }\) customers is:
   1. fewer than 20 ;
   2. more than 15 ;
   3. at least 12 but at most 24 .
3. The random variable \(W\) denotes the number of customers who pay by credit card from a random sample of \(\mathbf { 2 0 0 }\) customers. Calculate values for the mean and the variance of \(W\).
   [0pt] [3 marks]