Practice section 5 of 7

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]

(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]

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]

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]

(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]

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]

(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]

In a factory, a machine automatically cuts out metal caps. It is given that 0.5% of the caps are dented. Find the probability that out of 200 caps, at least 4 are dented.

A box containing 200 of such metal caps are analysed. The box is considered ‘substandard’ if it contains at least 4 dented caps. Calculate the probability that at most 6 out of 300 boxes will be ‘substandard’. [4]

A factory produces chocolate which are packed into boxes of 20 and delivered to shops for sale. A chocolate will not meet the minimum criteria for packing for sale if it weighs less than 20 grams. On average, 2% of the chocolate produced did not meet the minimum criteria.

(i) Find the probability that a randomly chosen box contain at least 1 chocolate that does not meet the minimum criteria. [2]

(ii) Find the probability that out of 4 randomly chosen boxes of chocolate, there are exactly 2 boxes with at least 1 chocolate that does not meet the minimum criteria. [2]