State general binomial conditions

3

1. State three conditions which must be satisfied for a situation to be modelled by a binomial distribution. George wants to invest some of his monthly salary. He invests a certain amount of this every month for 18 months. For each month there is a probability of 0.25 that he will buy shares in a large company, there is a probability of 0.15 that he will buy shares in a small company and there is a probability of 0.6 that he will invest in a savings account.
2. Find the probability that George will buy shares in a small company in at least 3 of these 18 months.

7

1. State two conditions which must be satisfied for a situation to be modelled by a binomial distribution. In a certain village 28\% of all cars are made by Ford.
2. 14 cars are chosen randomly in this village. Find the probability that fewer than 4 of these cars are made by Ford.
3. A random sample of 50 cars in the village is taken. Estimate, using a normal approximation, the probability that more than 18 cars are made by Ford.

6

1. State three conditions that must be satisfied for a situation to be modelled by a binomial distribution. On any day, there is a probability of 0.3 that Julie's train is late.
2. Nine days are chosen at random. Find the probability that Julie's train is late on more than 7 days or fewer than 2 days.
3. 90 days are chosen at random. Find the probability that Julie's train is late on more than 35 days or fewer than 27 days.

7 It is known that, on average, one match box in 10 contains fewer than 42 matches. Eight boxes are selected, and the number of boxes that contain fewer than 42 matches is denoted by \(Y\).

1. State two conditions needed to model \(Y\) by a binomial distribution. Assume now that a binomial model is valid.
2. Find
   (a) \(\mathrm { P } ( Y = 0 )\),
   (b) \(\mathrm { P } ( Y \geqslant 2 )\).
3. On Wednesday 8 boxes are selected, and on Thursday another 8 boxes are selected. Find the probability that on one of these days the number of boxes containing fewer than 42 matches is 0 , and that on the other day the number is 2 or more.

4. (a) State two conditions under which a random variable can be modelled by a binomial distribution. In the production of a certain electronic component it is found that \(10 \%\) are defective.
The component is produced in batches of 20 .
(b) Write down a suitable model for the distribution of defective components in a batch. Find the probability that a batch contains
(c) no defective components,
(d) more than 6 defective components.
(e) Find the mean and the variance of the defective components in a batch. A supplier buys 100 components. The supplier will receive a refund if there are more than 15 defective components.
(f) Using a suitable approximation, find the probability that the supplier will receive a refund.

3

1. Briefly describe the main features of a binomial distribution. I conduct an experiment by randomly selecting 10 cards, without replacement, from a normal pack of 52 .
2. Explain why the distribution of \(X\), the number of hearts obtained, is not \(\mathrm { B } \left( 10 , \frac { 1 } { 4 } \right)\).
   (2 marks)
   After making the appropriate adjustment to the experiment, which should be stated, so that the distribution is \(\mathrm { B } \left( 10 , \frac { 1 } { 4 } \right)\), find
3. the probability of getting no hearts,
4. the probability of getting 4 or more hearts.
5. If the modified experiment is repeated 50 times, find the total number of hearts that you would you expect to have drawn.