Verify conditions in context

5 Hebe attempts a crossword puzzle every day. The number of puzzles she completes in a week (7 days) is denoted by \(X\).

1. State two conditions that are required for \(X\) to have a binomial distribution.
   On average, Hebe completes 7 out of 10 of these puzzles.
2. Use a binomial distribution to find the probability that Hebe completes at least 5 puzzles in a week.
3. Use a binomial distribution to find the probability that, over the next 10 weeks, Hebe completes 4 or fewer puzzles in exactly 3 of the 10 weeks.

7 On average, \(25 \%\) of the packets of a certain kind of soup contain a voucher. Kim buys one packet of soup each week for 12 weeks. The number of vouchers she obtains is denoted by X .

1. State two conditions needed for X to be modelled by the distribution \(\mathrm { B } ( 12,0.25 )\). In the rest of this question you should assume that these conditions are satisfied.
2. Find \(\mathrm { P } ( \mathrm { X } \leqslant 6 )\). In order to claim a free gift, 7 vouchers are needed.
3. Find the probability that Kim will be able to claim a free gift at some time during the 12 weeks.
4. Find the probability that Kim will be able to claim a free gift in the 12th week but not before.

1. A bus company sells tickets for a journey from London to Oxford every Saturday. Past records show that \(5 \%\) of people who buy a ticket do not turn up for the journey.

The bus has seats for 48 people.
Each week the bus company sells tickets to exactly 50 people for the journey.
The random variable \(X\) represents the number of these people who do not turn up for the journey.

1. State one assumption required to model \(X\) as a binomial distribution. For this week's journey find,
2. 1. the probability that all 50 people turn up for the journey,
   2. \(\mathrm { P } ( X = 1 )\) The bus company receives \(\pounds 20\) for each ticket sold and all 50 tickets are sold. It must pay out \(\pounds 60\) to each person who buys a ticket and turns up for the journey but does not have a seat.
3. Find the bus company's expected total earnings per journey.

2. A fair coin is spun 6 times and the random variable \(T\) represents the number of tails obtained.

1. Give two reasons why a binomial model would be a suitable distribution for modelling \(T\).
2. Find \(\mathrm { P } ( T = 5 )\)
3. Find the probability of obtaining more tails than heads. A second coin is biased such that the probability of obtaining a head is \(\frac { 1 } { 4 }\) This second coin is spun 6 times.
4. Find the probability that, for the second coin, the number of heads obtained is greater than or equal to the number of tails obtained.

2. On average, \(25 \%\) of the packets of a certain kind of soup contain a voucher. Kim buys one packet of soup each week for 12 weeks. The number of vouchers she obtains is denoted by \(X\).

1. State two conditions needed for \(X\) to be modelled by the distribution \(\mathrm { B } ( 12,0.25 )\). In the rest of this question you should assume that these conditions are satisfied.
2. Find \(\mathrm { P } ( X \leqslant 6 )\). In order to claim a free gift, 7 vouchers are needed.
3. Find the probability that Kim will be able to claim a free gift at some time during the 12 weeks.
4. Find the probability that Kim will be able to claim a free gift in the 12th week but not before.
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9 A bag contains 100 black discs and 200 white discs. Paula takes five discs at random, without replacement. She notes the number \(X\) of these discs that are black.

1. Find \(\mathrm { P } ( X = 3 )\). Paula decides to use the binomial distribution as a model for the distribution of \(X\).
2. Explain why this model will give probabilities that are approximately, but not exactly, correct.
3. Paula uses the binomial model to find an approximate value for \(\mathrm { P } ( X = 3 )\). Calculate the percentage by which her answer will differ from the answer in part (ii). Paula now assumes that the binomial distribution is a good model for \(X\). She uses a computer simulation to generate 1000 values of \(X\). The number of times that \(X = 3\) occurs is denoted by \(Y\).
4. Calculate estimates of the limits between which two thirds of the values of \(Y\) will lie.

15 (e) State two necessary assumptions in context so that the distribution stated in part (a) is valid.

17 James is playing a mathematical game on his computer.
The probability that he wins is 0.6
As part of an online tournament, James plays the game 10 times.
Let \(Y\) be the number of games that James wins.
17

1. State two assumptions, in context, for \(Y\) to be modelled as \(B ( 10,0.6 )\)
   17
2. \(\quad\) Find \(\mathrm { P } ( Y = 4 )\)
   17
3. \(\quad\) Find \(\mathrm { P } ( Y \geq 4 )\)
   17
4. After practising the game, James claims that he has increased his probability of winning the game. In a random sample of 15 subsequent games, he wins 12 of them.
   Test a \(5 \%\) significance level whether James's claim is correct.
   \begin{center} \begin{tabular}{|l|l|l|l|} \hline \multicolumn{4}{|c|}{\begin{tabular}{l}

14

1. State two assumptions necessary for \(X\) to be modelled by a binomial distribution. \(\_\_\_\_\) [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
   14
2. Assume that \(X\) can be modelled by a binomial distribution. \(\_\_\_\_\)
   [0pt] [1 mark]
   \end{tabular}
   \hline & &
   \hline \end{tabular} \end{center} 14
3. (iii) Find \(\mathrm { P } ( X \geq 10 )\)
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4. In a random sample of 10 calls to a school, the number of calls which are complaints, \(Y\), may be modelled by a binomial distribution $$Y \sim \mathrm {~B} ( 10 , p )$$ The standard deviation of \(Y\) is 1.5 Calculate the possible values of \(p\).

12
12

1. 
   12
2. 
   12
3. 
   12
4. 
   \end{tabular} &

   It is known that, on average, \(40 \%\) of the drivers who take their driving test at a local test centre pass their driving test.

   Each day 32 drivers take their driving test at this centre.

   The number of drivers who pass their test on a particular day can be modelled by the distribution B (32, 0.4)

   State one assumption, in context, required for this distribution to be used.

   [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

   Find the probability that exactly 7 of the drivers on a particular day pass their test.

   [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

   Find the probability that, at most, 16 of the drivers on a particular day pass their test.

   [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

   Find the probability that more than 12 of the drivers on a particular day pass their test.

   [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

   
   \hline \end{tabular} \end{center}

   12

   Find the mean number of drivers per day who pass their test.

   [1 mark]

   12
5. Find the standard deviation of the number of drivers per day who pass their test.

1. In an experiment a group of children each repeatedly throw a dart at a target. For each child, the random variable \(H\) represents the number of times the dart hits the target in the first 10 throws.

Peta models \(H\) as \(\mathrm { B } ( 10,0.1 )\)

1. State two assumptions Peta needs to make to use her model.
2. Using Peta's model, find \(\mathrm { P } ( H \geqslant 4 )\) For each child the random variable \(F\) represents the number of the throw on which the dart first hits the target. Using Peta's assumptions about this experiment,
3. find \(\mathrm { P } ( F = 5 )\) Thomas assumes that in this experiment no child will need more than 10 throws for the dart to hit the target for the first time. He models \(\mathrm { P } ( F = n )\) as $$\mathrm { P } ( F = n ) = 0.01 + ( n - 1 ) \times \alpha$$ where \(\alpha\) is a constant.
4. Find the value of \(\alpha\)
5. Using Thomas' model, find \(\mathrm { P } ( F = 5 )\)
6. Explain how Peta's and Thomas' models differ in describing the probability that a dart hits the target in this experiment.