Review practice · Section 1 of 3

(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 marks

(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 marks

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

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 marks

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

Rose and Violet contest each other for the post of President of the Student Council of Floral Junior College. A pre-election survey indicates that Rose has the support of 45% of the school population while Violet has the support of the remaining students.

(i) Find the probability that Rose has fewer than 12 supporters in a group of 30 randomly selected students.3 marks

(ii) If the students are asked to queue up to cast their votes one at a time, find the probability that the 16th student is the 10th student who votes for Violet.3 marks

The probability that a particular make of light bulb is faulty is 0.2. The light bulbs are packed in boxes of 12.

(a) Find the probability that in a certain box, there are

(i) no faulty light bulbs

(ii) more than 3 faulty light bulbs

(b) A buyer accepts a consignment of 50 boxes, if, when he chooses two boxes at random, he finds that they contain no more than one faulty light bulb altogether. Find the probability that he will accept the consignment.

In a carton of apples, a sample of 8 apples is taken and examined for spoilt apples.

(i) State, in context, an assumption for the number of spoilt apples in the sample to be modelled by a binomial distribution.1 mark

The number of spoilt apples in a random sample of size 8 may be modelled by the distribution $B(8,p)$. If at least 2 apples in a sample are found to be spoilt, the carton is rejected. It is known that the probability of a carton being rejected is 0.04.

(ii) Write down an equation satisfied by $p$ and find the value of $p$.3 marks

(iii) 60 cartons of apples are loaded onto a lorry. Find the probability that more than 56 cartons loaded onto the lorry are not rejected.3 marks

A manufacturer produces pens in two colors, red and blue. On average, 35% of the pens produced are red pens.

(i) Find the probability that in a sample of 10 pens, more than half of them are red pens.2 marks

(ii) The manufacturer offers a special price on red pens by selling them in packs of six. Pens are selected at random to fill each pack. Given that five red pens have been selected, find the least value of $n$ such that the probability that at most $n$ more pens need to be selected to complete a pack of 6 red pens is greater than 0.98.4 marks