Practice section 6 of 7

It may be assumed that the dates of birth in a large population are distributed throughout the year so that the probability of a randomly chosen person’s date of birth being in any particular month may be taken as $\dfrac1{12}$.

(i) Find the probability that, of 6 people chosen at random, exactly 2 will have birthdays in January. [3]

(ii) Find the probability that, of 8 people chosen at random, at least one will have a birthday in January. [3]

(iii) $N$ people are chosen at random. Find the least value of $N$ such that the probability that at least one will have a birthday in January exceeds 0.9. [3]

(iv) Find the probability that, of 100 people chosen at random, at least 40 will have birthdays in May, June, July or Agusut. [4]

Temasek Bakery produces a large number of “lucky” cookies during the festive season. $p\%$ of these cookies contains a winning slip. The cookies are sold by packets. Each packet consists of 8 randomly chosen cookies and the mean number of cookies with a winning slip per packet is 3.2.

(i) Find the probability that a randomly chosen packet contains at most 4 cookies with a winning slip. [2]

(ii) Anand buys 12 packets of cookies. Find the probability that there are more than 10 packets that contain at most 4 cookies with a winning slip. [2]

(iii) Beng decides to buy $n$ packets of cookies. Find the least value of $n$ such that the probability that there is at least one packet that contain exactly 4 cookies with a winning slip is more than 0.90. [4]

According to the school rules, a student who arrives at school after 0730 hours is considered late.

(i) During the school term, a boy cycles to school on 5 days each week. On any given day, the probability that he arrives after 0730 hours is 0.1. For a period of 4 weeks, calculate the probability that he is late on at least one day but not more than 3 days. [2]

When the boy has cycled to school for the $n$th time, the probability that he arrived at school late at least once is greater than 0.99. Calculate the least value of $n$. [2]

(ii) A girl travels to school by bus on 5 days a week. Over a long period of time, the variance of the number of days per week on which she is late is 0.8. Given that $p$ denotes the probability that she arrives late and that $p<0.5$, evaluate $p$. [3]

Samples, each of 12 articles, are taken at random from a large consignment in which 25% of articles are defective.

Find the number of defective articles which is most likely to occur in a single sample, and find the probability of obtaining this number.

If 100 samples of 12 articles are to be examined, calculate the number of samples in which you would expect to find 4 or more defective articles.

One question in a survey is as follows:

“Do you use Facebook during office hours at least once a day? Yes or No?”

It is known that on average, 7 out of 9 staff members answered “yes” to the above question.

For a random sample of 60 staff members,

(i) find the standard deviation of the number of staff members who answer “yes”, [1]

(ii) find the probability that the number of staff members who answer “yes” exceeds the mean, [3]

(iii) find the most probable number of staff members who answer “yes”, [2]

(iv) state, with a reason, if the above answers are valid if the total number of staff members in the company is small. [1]

The random variable $X$ is the number of successes in $n$ independent trials of an experiment in which the probability of a success at any trial is $p$.

Denoting $P(X=k)$ by $p_k$, it is given that $\dfrac{p_k}{p_{k-1}}=\dfrac{(n-k+1)p}{k(1-p)}$, $k=1,2,\ldots,n$.

A distribution is said to be bimodal if it has two modes.

Find the least value of $n$, and the corresponding modes of $X$, given that $X$ is bimodal and that $p=\dfrac{18}{25}$. [3]