Discussion questions · Section 1 of 2

From past records, the purchases at a fast food outlet show that 63% of customers buy a burger. This figure is made up of 30% who buy a beef burger, 24% who buy a fish burger and 9% who buy a chicken burger. No one buys more than one burger. Find the probability that, in a random sample of 27 customers,

(i) at least 23 customers buy a burger,

(ii) less than 26 customers do not buy a fish burger,

(iii) more than 15 customers buy either a beef or a chicken burger,

(iv) no less than 8 customers do not buy a burger,

(v) exactly 4 customers buy a chicken burger given that fewer than 6 customers buy a chicken burger.

Customers at a supermarket pay for their purchases either by cash or credit card. The probability of a randomly chosen customer paying by credit card is 0.3. Find the probability that, on a randomly chosen day, there are more than 6 customers among the first 20 customers who pay by credit card and the fifth customer is the second customer who does so.

After the Monetary Authority of Singapore’s report on the structured financial products sold by financial institutions in Singapore, a survey found that 95% of the respondents would favour greater consumer protection.

(i) Find the probability that out of 30 respondents, 25 of them would favour greater consumer protection.

(ii) Find the least number of respondents surveyed such that the probability that at least 40 of them would favour greater consumer protection exceeds 0.980.

On average, 30% of the patients having a certain medical treatment experience severe headache. A random sample of 12 patients is given the treatment.

(i) Write down the distribution of the number of patients who experience severe headache, and state an assumption for the choice of the distribution.

(ii) Find the probability that fewer than 3 patients experience severe headache.

(iii) Five random samples of 12 patients are given the treatment. Find the probability that two of these five samples have fewer than 3 patients who experience severe headache.

The random variable $X\sim B(16,p)$ where $p<0.5$. If the variance of $X$ is 3.36, find the value of $p$. Find also the probability that $X$ is less than the mean.

In each batch of manufactured articles the proportion of defective articles is $p$. From each batch a random sample of nine is taken and each of the nine articles is examined. If two or more of the nine articles are found to be defective that batch is rejected; otherwise it is accepted. Prove that the probability that a batch is accepted is $(1-p)^8(1+8p)$.

It is decided to modify the sampling scheme so that, when one defective is found in the sample, a second sample of nine is taken and the batch rejected if this contains any defectives. With this exception, the original scheme is continued. Find an expression in terms of $p$ for the probability that a batch is accepted.

For this modified scheme evaluate the average number sampled per manufactured batch over a large number of batches when $p$ has the value 0.1.