Question

A. find the probability of getting exactly 6 girls in 8 births.

b. find the probability of getting 6 or more girls in 8 births.
c. which probability is relevant for determining whether 6 is a significantly girls in 8 births: the result from part (a) or part (b)

Answer 1

Assuming each birth can give life to a boy or girl with equal chance, then the probability of having a girl is
`(1)/(2)`

The probability of having exactly six girls is given by Bernoulli rule:
`\binom{8}{6}\left((1)/(2)\right)^6\left((1)/(2)\right)^2 = (8!)/(6!2!)\left((1)/(2)\right)^8 = (8\cdot 7)/(2) (1)/(2^8) = (28)/(2^8) \approx 0.11`

As for the second point, let's compute the probabilty of having zero, one or two boys, which is the same of having six or more girls, but requires less calculations:

`\binom{8}{0} \left((1)/(2)\right)^8 \left((1)/(2)\right)^0 + \binom{8}{1} \left((1)/(2)\right)^7 \left((1)/(2)\right)^1 + \binom{8}{2} \left((1)/(2)\right)^6 \left((1)/(2)\right)^2 = \left(\binom{8}{0} + \binom{8}{1} + \binom{8}{2}\right) (1)/(2^8)`

which simplifies to

`((1+8+28))/(2^8) = (37)/(2^8) \approx 0.14`

Finally, I am not sure of what you mean with the point c, so please clarify the question.