Question

Suppose that 50% of all babies born in a particular hospital are girls. If 6 babies born in the hospital are randomly selected, what is the probability that fewer than 3 of them are girls?

Answer 1

Answer on question #44689 - Math - Statistics and Probability

Suppose that 50% of all babies born in a particular hospital are girls. If 6 babies born in the hospital are randomly selected, what is the probability that fewer than 3 of them are girls?

Solution

It's an example of a binomial experiment.

Let $p$ - probability that baby is a girl, $q$ - probability that baby is a boy. So:

$p = 0.5$ - baby is a girl.

$q = 1 - 0.5 = 0.5$ - baby is a boy.

We have 6 trials. To calculate the probability "fewer than 3", we need to add probabilities that 0, 1 and 2 babies are girls:

$$P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)$$ $$P(X = 0) = \binom{6}{0} \cdot p^0 \cdot q^6 = 0.015625$$ $$P(X = 1) = \binom{6}{1} \cdot p^1 \cdot q^5 = 0.09375$$ $$P(X = 2) = \binom{6}{2} \cdot p^2 \cdot q^4 = 0.234375$$

So answer is:

$$P(X < 3) = 0.015625 + 0.09375 + 0.234375 = 0.34375$$

Answer

$$P(X < 3) = 0.34375.$$

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