(a) What is the probability that exactly 3 of the surveyed customers like the Saturday option?
(b) What is the probability that exactly 5 of these customers like the Saturday option?
(c) What is the probability that at most 5 customers in the sample like the Saturday option?
(d) What is the probability that at least 5 of them like the Saturday option?
Answer :
Given :
Probability of success $(p) = 0.83$
Sample size $(n) = 8$
Solution :
A) The probability that exactly 3 of the surveyed customers like the Saturday option :
$$P(x = 3) = \binom{n}{x} \cdot p^x \cdot (1-p)^{n-x}$$ $$= \binom{8}{3} \cdot 0.83^3 \cdot (1 – 0.83)^{8-3}$$ $$= 0.0045$$
B) The probability that exactly 5 of these customers like the Saturday option :
$$P(x = 5) = \binom{n}{x} \cdot p^x \cdot (1-p)^{n-x}$$ $$= \binom{8}{5} \cdot 0.83^5 \cdot (1 – 0.83)^{8-5}$$ $$= 0.1084$$
C) The probability that at most 5 customers in the sample like the Saturday option :
$$P(x \leq 5) = \sum_{0}^{5} \binom{n}{x} \cdot p^x \cdot (1-p)^{n-x}$$ $$= \sum_{0}^{5} \binom{8}{x} \cdot 0.83^x \cdot (1 – 0.83)^{8-x}$$ $$= 0.1412$$
D) The probability that at least 5 of them like the Saturday option :
$$P(x \geq 5) = \sum_{5}^{8} \binom{n}{x} \cdot p^x \cdot (1-p)^{n-x}$$ $$= \sum_{5}^{8} \binom{8}{x} \cdot 0.83^x \cdot (1 – 0.83)^{8-x}$$ $$= 0.9672$$
