Question

Assuming that half the population is vegetarian so that the chance of an individual being a vegetation is ½ and assuming that 100 investigators can take a sample of 10 individuals to see whether they are vegetarian, how many investigators would you expect to report that three people or less were vegetarians?

Answer 1

Let's find probability that a random sample of 10 individuals contain 3 or less vegetarians. Denote number of vegetarians in the sample by 10. Then $\xi$ has Binomial distribution with parameters $n = 10$ and $p = 0.5$. Thus

$$\begin{array}{l} P(\xi \leq 3) = P(\xi = 0) + P(\xi = 1) + P(\xi = 2) + P(\xi = 3) \\ = \binom{10}{0} 0.5^{0} 0.5^{10} + \binom{10}{1} 0.5^{1} 0.5^{9} + \binom{10}{2} 0.5^{2} 0.5^{8} + \binom{10}{3} 0.5^{3} 0.5^{7} \\ = 0.5^{10} \left( \binom{10}{0} + \binom{10}{1} + \binom{10}{2} + \binom{10}{3} \right) \\ = 0.5^{10} (1 + 10 + 45 + 120) = \frac{176}{1024} = \frac{11}{64} \end{array}$$

Thus expected number of investigators reporting about 3 or less vegetarians equals

$$100 \cdot \frac{11}{64} = \frac{275}{16} = 17.1875$$