Probability Calculation in a Smoker Population

In a city where 10% of the population are smokers, we will calculate the probability of various scenarios when a random sample of 5 people is drawn. We will use the binomial distribution to model the number of smokers in the sample. The binomial distribution is given by:

1. None are Smokers

Let X be the number of smokers in the sample. The probability that none of the 5 people are smokers can be calculated using the binomial formula:

P(X 0) sum_{r0}^{0} binom{5}{r} (0.1)^r (0.9)^{5-r}

Since the term for r 0 is the only term, the calculation simplifies to:

P(X  0)  binom{5}{0} (0.1)^0 (0.9)^5  1 * 1 * 0.59049  0.59049 approx 0.59

2. At Least One is a Smoker

To find the probability that at least one person in the sample is a smoker, we can subtract the probability of none being smokers from 1:

P(X geq 1)  1 - P(X  0)  1 - 0.59049  0.40951 approx 0.41

3. Four are Smokers

The probability that exactly four out of the five people in the sample are smokers is given by:

P(X  4)  binom{5}{4} (0.1)^4 (0.9)^1  5 * (0.0001) * 0.9  4.5 * 10^{-4} approx 0.00045

4. Between 1 and 3 Both Inclusive are Smokers

The probability that between one and three (inclusive) people in the sample are smokers can be calculated by summing the probabilities of 1, 2, and 3 smokers:

P(1 leq X leq 3)  P(X  1)   P(X  2)   P(X  3)

These probabilities can be calculated as follows:

P(X  1)  binom{5}{1} (0.1)^1 (0.9)^4  5 * 0.1 * 0.6561  0.32805

P(X  2)  binom{5}{2} (0.1)^2 (0.9)^3  10 * 0.01 * 0.729  0.0729

P(X  3)  binom{5}{3} (0.1)^3 (0.9)^2  10 * 0.001 * 0.81  0.0081

Summing these probabilities:

P(1 leq X leq 3)  0.32805   0.0729   0.0081  0.40905 approx 0.409

Conclusion

In summary, using the binomial distribution, we calculated the probabilities of various scenarios for a random sample of 5 people in a city where 10% of the population are smokers. These calculations are fundamental in understanding the distribution of characteristics within a population and can be applied in various fields, including public health, social science, and user behavior analysis.