Finding binomial parameters from properties

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 \(k\).

3 An experiment consists of throwing a biased die 30 times and noting the number of 4 s obtained. This experiment was repeated many times and the average number of 4 s obtained in 30 throws was found to be 6.21.

1. Estimate the probability of throwing a 4.
   ..................................................................................................................................... .
   \section*{Hence}
2. find the variance of the number of 4 s obtained in 30 throws,
3. find the probability that in 15 throws the number of 4 s obtained is 2 or more.

9 A variable \(X\) has the distribution \(\mathrm { B } ( 11 , p )\).

1. Given that \(p = \frac { 3 } { 4 }\), find \(\mathrm { P } ( X = 5 )\).
2. Given that \(\mathrm { P } ( X = 0 ) = 0.05\), find \(p\).
3. Given that \(\operatorname { Var } ( X ) = 1.76\), find the two possible values of \(p\).

7

1. Andrew plays 10 tennis matches. In each match he either wins or loses.
   1. State, in this context, two conditions needed for a binomial distribution to arise.
   2. Assuming these conditions are satisfied, define a variable in this context which has a binomial distribution.
   3. The random variable \(X\) has the distribution \(\mathrm { B } ( 21 , p )\), where \(0 < p < 1\). Given that \(\mathrm { P } ( X = 10 ) = \mathrm { P } ( X = 9 )\), find the value of \(p\).

8

1. A biased coin is thrown twice. The probability that it shows heads both times is 0.04 . Find the probability that it shows tails both times.
2. A nother coin is biased so that the probability that it shows heads on any throw is p . The probability that the coin shows heads exactly once in two throws is 0.42 . Find the two possible values of p.

1. Some children are playing a game involving throwing a ball into a bucket. Each child has 3 throws and the number of times the ball lands in the bucket, \(x\), is recorded. Their results are given in the table below.

1. Find \(\bar { x }\) (1) Sandra decides to model the game by assuming that on each throw, the probability of the ball landing in the bucket is 0.4 for every child on every throw and that the throws are all independent. The random variable \(S\) represents the number of times the ball lands in the bucket for a randomly selected child.
2. Find \(\mathrm { P } ( S = 2 )\)
3. Complete the table below to show the probability distribution for \(S\).

   \(s\)	0	1	2	3
   \(\mathrm { P } ( S = s )\)		0.432		0.064

   Ting believes that the probability of the ball landing in the bucket is not the same for each throw. He suggests that the probability will increase with each throw and uses the model $$p _ { i } = 0.15 i + 0.10$$ where \(i = 1,2,3\) and \(p _ { i }\) is the probability that the \(i\) th throw of the ball, by any particular child, will land in the bucket.
   The random variable \(T\) represents the number of times the ball lands in the bucket for a randomly selected child using Ting's model.
4. Show that
   1. \(\mathrm { P } ( T = 3 ) = 0.055\)
   2. \(\mathrm { P } ( T = 1 ) = 0.45\) (5)
5. Complete the table below to show the probability distribution for \(T\), stating the exact probabilities in each case.

   \(t\)	0	1	2	3
   \(\mathrm { P } ( T = t )\)		0.45		0.055

6. State, giving your reasons, whether Sandra's model or Ting's model is the more appropriate for modelling this game.

12 A manufacturer of steel rods checks the length of each rod in randomly selected batches of 10 rods. 100 batches of 10 rods are checked and \(x\), the number of rods in each batch which are too long, is recorded. Summary statistics are as follows. \(n = 100\) $$\sum x = 210 \quad \sum x ^ { 2 } = 604$$

1. Calculate
   Layla decides to use a binomial distribution to model the number of rods which are too long in a batch of 10 .
2. Write down the parameters that Layla should use in her model.
3. Use Layla's model to determine the expected number of batches out of 100 in which there are exactly 2 rods which are too long.

4 Two random variables \(X\) and \(Y\) have the distributions \(\mathrm { B } ( m , p )\) and \(\mathrm { B } ( n , p )\) respectively, where \(p > 0\). It is known that

• \(\mathrm { E } ( Y ) = 2 \mathrm { E } ( X )\)
• \(\operatorname { Var } ( Y ) = 1.2 \mathrm { E } ( X )\).

Determine the value of \(p\).

1. A call centre routes incoming telephone calls to agents who have specialist knowledge to deal with the call. The probability of a caller, chosen at random, being connected to the wrong agent is p.

The probability of at least 1 call in 5 consecutive calls being connected to the wrong agent is 0.049 The call centre receives 1000 calls each day.

1. Find the mean and variance of the number of wrongly connected calls a day.
2. Use a Poisson approximation to find, to 3 decimal places, the probability that more than 6 calls each day are connected to the wrong agent.
3. Explain why the approximation used in part (b) is valid. The probability that more than 6 calls each day are connected to the wrong agent using the binomial distribution is 0.8711 to 4 decimal places.
4. Comment on the accuracy of your answer in part (b).

6 In this question you must show detailed reasoning. The random variable \(T\) has a binomial distribution. It is known that \(\mathrm { E } ( T ) = 5.625\) and the standard deviation of \(T\) is 1.875 . Find the values of the parameters of the distribution.

The random variable \(X\) is such that \(X \sim B\left(n, \frac{1}{3}\right)\) The standard deviation of \(X\) is 4 Find the value of \(n\). Circle your answer. [1 mark] 9 \quad 12 \quad 18 \quad 72

The discrete random variables \(X\) and \(Y\) can be modelled by the distributions $$X \sim \text{B}(40, p)$$ $$Y \sim \text{B}(25, 0.6)$$ It is given that the mean of \(X\) is equal to the variance of \(Y\)

1. Find the value of \(p\) [3 marks]
2. Find P(\(Y = 17\)) [1 mark]

The random variable \(X\) is such that \(X \sim B(n, p)\) The mean value of \(X\) is 225 The variance of \(X\) is 144 Find \(p\). Circle your answer. [1 mark] 0.36 \quad 0.6 \quad 0.64 \quad 0.8

A customer service centre records every call they receive. It is found that 30% of all calls made to this centre are complaints. A sample of 20 calls is selected. The number of calls in the sample which are complaints is denoted by the random variable \(X\).

1. State two assumptions necessary for \(X\) to be modelled by a binomial distribution. [2 marks]
2. Assume that \(X\) can be modelled by a binomial distribution.
   1. Find P(\(X = 1\)) [1 mark]
   2. Find P(\(X < 4\)) [2 marks]
   3. Find P(\(X \geq 10\)) [2 marks]
3. 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 $$Y \sim \text{B}(10, p)$$ The standard deviation of \(Y\) is 1.5 Calculate the possible values of \(p\). [3 marks]

In this question you must show detailed reasoning. The random variable \(T\) has a binomial distribution. It is known that \(E(T) = 5.625\) and the standard deviation of \(T\) is \(1.875\). Find the values of the parameters of the distribution. [5]