Question

A nationwide survey of college seniors by the
University of Michigan revealed that almost 70% disapprove
of daily pot smoking, according to a report in
Parade. If 12 seniors are selected at random and asked
their opinion, find the probability that the number who
disapprove of smoking pot daily is
(a) anywhere from 7 to 9;
(b) at most 5;
(c) not less than 8.

Answer 1

Question #72633, Math / Statistics and Probability

Solution: Let $X$ be the random variable denoting number of college seniors who disapprove of smoking pot daily from the sample of 12.

So, clearly $X \sim$ Binomial (12, 0.7).

The probability that the number of people who disapprove of smoking pot daily is

a) Anywhere from 7 to 9

Required probability is $P(X=7) + P(X=8) + P(X=9)$

$$= {}^{12}\mathrm{C}_7 \cdot (0.7)^7 \cdot (0.3)^5 + {}^{12}\mathrm{C}_8 \cdot (0.7)^8 \cdot (0.3)^4 + {}^{12}\mathrm{C}_9 \cdot (0.7)^9 \cdot (0.3)^3.$$

b) At most 5

Required probability is $P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)$

$$= (0.3)^{12} + {}^{12}\mathrm{C}_1 \cdot (0.7)^* (0.3)^{11} + {}^{12}\mathrm{C}_2 \cdot (0.7)^2 \cdot (0.3)^{10} + {}^{12}\mathrm{C}_3 \cdot (0.7)^3 \cdot (0.3)^9$$ $$+ {}^{12}\mathrm{C}_4 \cdot (0.7)^4 \cdot (0.3)^8 + {}^{12}\mathrm{C}_5 \cdot (0.7)^5 \cdot (0.3)^7.$$

c) Not less than 8

Required probability is $P(X=12) + P(X=11) + P(X=10) + P(X=9) + P(X=8)$

$$= (0.7)^{12} + {}^{12}\mathrm{C}_{11} \cdot (0.3)^* (0.7)^{11} + {}^{12}\mathrm{C}_{10} \cdot (0.3)^2 \cdot (0.7)^{10} + {}^{12}\mathrm{C}_9 \cdot (0.3)^3 \cdot (0.7)^9$$ $$+ {}^{12}\mathrm{C}_8 \cdot (0.3)^4 \cdot (0.7)^8.$$

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