## 34. Family Breakfast

Every Sunday, a married couple has breakfast with their mothers. Unfortunately, each spouse's relationship with their mother-in-law is quite strained: both know that there are two out of three chances of getting into an argument with their mother-in-law upon meeting. In the event of a conflict, the other spouse takes the side of their own mother (and thus argues with their spouse) about half the time; equally often, they defend their spouse (or husband) and argue with their mother.

Assuming that the arguments between each spouse and their mother-in-law are independent of each other, what do you think is the proportion of Sundays when the spouses manage to avoid arguing with each other?
34. Two out of three times, the husband argues with his mother-in-law, and one out of three times (since $1 / 2 \cdot 2 / 3=1 / 3$) this leads to a family quarrel.

Two out of three times, the wife argues with her mother-in-law, and one out of three times this leads to a family quarrel.

The proportion of Sundays when a family quarrel occurs twice - both due to the husband's arguments with his mother-in-law and the wife's arguments with her mother-in-law - is $1 / 3 \cdot 1 / 3=1 / 9$, since these arguments are independent of each other. Therefore, the proportion of Sundays when a quarrel occurs between the spouses for at least one reason is

$$
1 / 3+1 / 3-1 / 9=5 / 9
$$

Thus, the proportion of Sundays that pass without such family quarrels is only

$$
1-5 / 9=4 / 9
$$