Probability of having less than 3 females if....

Question

Assuming that half the population is female and assuming that 100 samples of 10 individuals are taken, how many samples would you expect to have 3 or less females?

Can someone please explain how this is done?

Answer 1

For a fixed sample of $10$ individuals, assuming a large population, the probability $p_F$ of $3$ or less females is $$ p_F = \sum_{i=0}^3 \binom{10}{i} \left(\frac{1}{2}\right)^{10} = \left(\frac{1}{2}\right)^{10}(1 + 10 + 45 + 120) = \frac{176}{1024}. $$ If you perform this experiment $100$ times you therefore expect to see fewer than $3$ females $$ 100p_F = \frac{17600}{1024} \approx 17.1875 $$ times.

Answer 2

The probability that a sample has $3$ or less females is $$p = \biggl(\frac{1}{2}\biggr)^{10} + 10 \cdot \biggl(\frac{1}{2}\biggr)^{10} + \binom{10}{2}\biggl(\frac{1}{2}\biggr)^{10} + \binom{10}{3}\biggl(\frac{1}{2}\biggr)^{10}$$

Thus we have to calculate the expected value of a binomial random variable with the previous $p$ , i.e. $$N = 100 \cdot p$$

Answer 3

I hope this may be of any help