Practice section 4 of 7

Teams of 8 people enter a shooting competition. Each member of a team takes a shot at the target. The number of points scored by a team is obtained by $2x$ where $x$ is the number of shots that hit the target. For a particular team, each member of the team independently has a probability of 0.75 of hitting the target.

(i) Find the probability that a randomly chosen team scores more than 6 points.

(ii) Find the number of points that a team is most likely to score.

A bag contains 4 red, 5 blue and 6 green balls. The balls are indistinguishable except for their colour. A trial consists of drawing a ball at random from the bag, noting its colour and replacing it in the bag. A game is played by performing 10 trials in all.

The player receives $0.50 for every blue ball drawn in a game. He receives nothing if balls of other colours are drawn.

(a) Find the probability that, in a game, he receives

(i) exactly $3,

(ii) more than $2.50,

(b) Find also the amount a player is most likely to receive.

(c) At the start of a tournament, each player plays the above game once. Players who earned more than $ $k$ proceed to the next round. Find the least value of $k$ such that, in a random sample of 10 players, the probability that all the 10 players proceed to the next round is less than 0.1.

A manufacturer produces a large number of coloured glowing sticks, which are packed in packets of 8. Each packet consists of randomly chosen coloured glowing sticks. On average, 12% of the glowing sticks are green.

(i) Explain the significance of ‘large number’ in the first sentence.

(ii) Find the probability that a randomly chosen packet of glowing sticks contains at least 3 green glowing sticks.

(iii) Ian buys 40 randomly chosen packets of glowing sticks for a concert. Find the probability that fewer than 6 of these packets contain at least 3 green glowing sticks.

On average, the proportion of glowing sticks that are blue is $p$. It is known that the modal number of blue glowing sticks in a packet is 3.

(iv) Use this information to find exactly the range of values that $p$ can take.

A company manufactures a wide variety of components for use in domestic appliances.

The company introduces a new type of light fitting for refrigerators. Each day a supervisor selects a sample of 100 of the light fittings for testing.

(i) How should the light fittings be selected? Give a reason for this method of selection.

The supervisor records the number of faulty light fittings found on each of 150 working days. Her results are shown in the table.

Number faulty	Number of days
0	4
1	19
2	38
3	41
4	22
5	16
6	6
7	4
8 or more	0

(ii) Use the information in the table to estimate $p$, the probability that a light fitting is faulty.

(iii) Assuming the number of faulty fittings found each day follows the binomial distribution $B(100,p)$, find the expected number of days on which 3 faulty fittings are found in a period of 150 working days.

(i) In a small plot of land, there are 20 seeds being sown, of which only 4 will germinate. If 5 seeds are selected at random without replacement, find the probability that at least a seed that germinates would be selected. [3]

(ii) In a huge plot of land where a large number of seeds are sown, 10% of the seeds would germinate. If 5 seeds are selected at random without replacement, find the probability that more than a seed that germinates would be selected. [3]

State an assumption made in calculating the probability in (ii). [1]

When a large number of flashlights leaving a factory is inspected, it is found that the bulb is faulty in 1% of the flashlights and the switch is faulty in 1.5% of them. Assuming that the faults occur independently and at random, find the probability that a sample of

(i) 10 flashlights contains no faulty bulb, [2]

(ii) 80 flashlights contains at least one flashlight with both a faulty bulb and a faulty switch. [3]