2022 ACJC P2 Q11
(a)
(i) It is known that the probability of a customer using e-payment at a hawker stall is $0.25$. A group of customers is chosen at random, find the probability that the ${{7}^{\text{th}}}$ chosen customer is the ${{5}^{\text{th}}}$ customer using e-payment at the hawker stall.
[3]
(ii) A sample of $40$ customers were randomly chosen from the hawker stall each day. In a month of $30$ days, find the probability that there is at least $15$ days with at most $10$ customers per sample making e-payment.
[3]
(b)
The probability that a hawker uses the online delivery platform Foodgowhere is $p$. A random sample of $~n$ hawkers is taken and the random variable $X$ denotes the number of hawkers in the sample that uses Foodgowhere.
(i) Explain what is meant by a random sample in this context.
[1]
Assuming that $X$ follows a binomial distribution.
(ii) It is given that $\text{P}\left( X\le 1 \right)=\text{0}\text{.05303}$ and the expected number of hawkers using Foodgowhere is $3.96$. Write down two equations satisfied by $p$ and $~n$. Hence find the value of $p$ and $~n$.
[3]
(iii) Given that $n=15$, find the set of values of $p$ so that that the most likely number of hawkers in the sample who uses Foodgowehere is $5$.
[3]
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