Washing up by lottery

Four housemates draw one name from a hat every week to decide who washes up. This week one of them complained: the washing up has been mine three weeks in a row!

$4$ housemates; each week one name is drawn from a hat. One complained: $3$ weeks in a row!

$n=4$ people, one name a week. Complaint: $3$ weeks running.

Each week each person has a one in four chance. For one named person to be drawn three weeks running, the chance is one quarter times one quarter times one quarter, which is one in sixty-four.

Each week a person's chance is $p=\frac{1}{4}$. Three weeks running for one named person: $p^3=\frac{1}{64}$.

Chance $p=\frac{1}{4}$; $p^3=\frac{1}{64}$.

But the unlucky person could have been any of the four. Four times one in sixty-four is one in sixteen, about six per cent. Rare, but not scandalous.

But it could have been any of the $4$: $4\times \frac{1}{64}=\frac{1}{16}=0.0625$, about $6\%$.

Anyone: $4p^3=\frac{1}{16}\approx 0.06$.

A fair lottery does not promise that the chores spread evenly every week; it only gives everyone the same chance. Over forty weeks, each person is expected to wash up ten times.

A lottery gives equal chances, not an equal share each week. In $40$ weeks, $40\times \frac{1}{4}=10$ turns each.

$40$ weeks: $40p=10$ turns each.