Combined probability with other distributions

7 In the region of Arka, the total number of households in the three villages Reeta, Shan and Teber is 800 . Each of the households was asked about the quality of their broadband service. Their responses are summarised in the following table.

1. 1. Find the probability that a randomly chosen household is in Shan and has poor broadband service.
   2. Find the probability that a randomly chosen household has good broadband service given that the household is in Shan.
      In the whole of Arka there are a large number of households. A survey showed that \(35 \%\) of households in Arka have no broadband service.
2. 1. 10 households in Arka are chosen at random. Find the probability that fewer than 3 of these households have no broadband service.
   2. 120 households in Arka are chosen at random. Use an approximation to find the probability that more than 32 of these households have no broadband service.
      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 Company \(A\) produces bags of sugar. An inspector finds that on average \(10 \%\) of the bags are underweight. 10 of the bags are chosen at random.

1. Find the probability that fewer than 3 of these bags are underweight.
   The weights of the bags of sugar produced by company \(B\) are normally distributed with mean 1.04 kg and standard deviation 0.06 kg .
2. Find the probability that a randomly chosen bag produced by company \(B\) weighs more than 1.11 kg .
   \(81 \%\) of the bags of sugar produced by company \(B\) weigh less than \(w \mathrm {~kg}\).
3. Find the value of \(w\).

3 A company produces small boxes of sweets that contain 5 jellies and 3 chocolates. Jemeel chooses 3 sweets at random from a box.

1. Draw up the probability distribution table for the number of jellies that Jemeel chooses.
   The company also produces large boxes of sweets. For any large box, the probability that it contains more jellies than chocolates is 0.64 . 10 large boxes are chosen at random.
2. Find the probability that no more than 7 of these boxes contain more jellies than chocolates.

2 The faces of a biased die are numbered \(1,2,3,4,5\) and 6 . The random variable \(X\) is the score when the die is thrown. The following is the probability distribution table for \(X\). The die is thrown 3 times. Find the probability that the score is 4 on not more than 1 of the 3 throws.

3 Jake attempts the crossword puzzle in his daily newspaper every day. The probability that he will complete the puzzle on any given day is 0.75 , independently of all other days.

1. Find the probability that he will complete the puzzle at least three times over a period of five days.
   Kenny also attempts the puzzle every day. The probability that he will complete the puzzle on a Monday is 0.8 . The probability that he will complete it on a Tuesday is 0.9 if he completed it on the previous day and 0.6 if he did not complete it on the previous day.
2. Find the probability that Kenny will complete the puzzle on at least one of the two days Monday and Tuesday in a randomly chosen week.

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.

1. At a cafe, customers ordering hot drinks order either tea or coffee.

Of all customers ordering hot drinks, \(80 \%\) order tea and \(20 \%\) order coffee. Of those who order tea, \(35 \%\) take sugar and of those who order coffee \(60 \%\) take sugar.

1. A random sample of 12 customers ordering hot drinks is selected. Find the probability that fewer than 3 of these customers order coffee.
2. 1. A randomly selected customer who orders a hot drink is chosen. Show that the probability that the customer takes sugar is 0.4
   2. Write down the distribution for the number of customers who take sugar from a random sample of \(n\) customers ordering hot drinks.
3. A random sample of 10 customers ordering hot drinks is selected.
   1. Find the probability that exactly 4 of these 10 customers take sugar.
   2. Given that at least 3 of these 10 customers take sugar, find the probability that no more than 6 of these 10 customers take sugar.
4. In a random sample of 150 customers ordering hot drinks, find, using a suitable approximation, the probability that at least half of them take sugar.

3 An auction house offers items of jewellery for sale at its public auctions. Each item has a reserve price which is less than the lower price estimate which, in turn, is less than the upper price estimate. The outcome for any item is independent of the outcomes for all other items. The auction house has found, from past records, the following probabilities for the outcomes of items of jewellery offered for sale.

1. The probability of winning a prize when playing a single game of Pento is \(\frac { 1 } { 5 }\)

When more than one game is played the games are independent.
Sam plays 20 games.

1. Find the probability that Sam wins 4 or more prizes. Tessa plays a series of games.
2. Find the probability that Tessa wins her 4th prize on her 20th game. Rama invites Sam and Tessa to play some new games of Pento. They must pay Rama \(\pounds 1\) for each game they play but Rama will pay them \(\pounds 2\) for the first time they win a prize, \(\pounds 4\) for the second time and \(\pounds ( 2 w )\) when they win their \(w\) th prize ( \(w > 2\) ) Sam decides to play \(n\) games of Pento with Rama.
3. Show that Sam's expected profit is \(\pounds \frac { 1 } { 25 } \left( n ^ { 2 } - 16 n \right)\) Given that Sam chose \(n = 15\)
4. find the probability that Sam does not make a loss. Tessa agrees to play Pento with Rama. She will play games until she wins \(r\) prizes and then she will stop.
5. Find, in terms of \(r\), Tessa's expected profit.