Let's assume that the birth of a girl and a boy is equally likely. It is known that a certain family has two children.

a) What is the probability that one is a boy and one is a girl?

b) It is additionally known that one of the children is a boy. What is the probability now that there is one boy and one girl in the family?

c) It is additionally known that the boy was born on a Monday. What is the probability now that there is one boy and one girl in the family?
a) Children appear in a certain sequence (MM, MD, DM, or DD). All sequences are equally likely, and the probability of each is 1/4. The condition "boy and girl" is favorable for two outcomes MD and DM, so the probability of this is $2: 4 = 1 / 2$.

b) Out of the four sequences, three remain equally likely: MM, MD, and DM. Therefore, the probability of the event "boy and girl" is now $2 / 3$.

c) Let's introduce the days of the week: $\mathrm{M}_{M}$ - a boy born on Monday, $\mathrm{M}_{O}$ - a boy born on another day of the week. According to the condition, five sequences are possible: $\mathrm{M}_{M} \mathrm{M}_{M}, \mathrm{M}_{M} \mathrm{M}_{O}$, $\mathrm{M}_{O} \mathrm{M}_{M}, \mathrm{M}_{M} D, D \mathrm{M}_{M}$

Considering that the probability of being born on a Monday is $1 / 7$, and on another day is $6 / 7$, we find that the total probability of the possible sequences is

$\frac{1}{2} \cdot \frac{1}{7} \cdot \frac{1}{2} \cdot \frac{1}{7} + \frac{1}{2} \cdot \frac{1}{7} \cdot \frac{1}{2} \cdot \frac{6}{7} + \frac{1}{2} \cdot \frac{6}{7} \cdot \frac{1}{2} \cdot \frac{1}{7} + \frac{1}{2} \cdot \frac{1}{7} \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{7} = \frac{1}{4} \cdot \frac{27}{49}$.

Of these sequences, only two: $\mathrm{M}_{M} D$ and $D \mathrm{M}_{M}$ favor the event "boy and girl". Their total probability is $1 / 4 \cdot 2 / 7$.

Therefore, the conditional probability of the event "boy and girl" given the condition "there is a boy born on Monday" is $2 / 7 : 27 / 49 = 14 / 27$.

## Answer

a) $1 / 2 ;$ b) $2 / 3 ;$ c) $14 / 27$

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