Problem — The paradox of the two children born on a Tuesday

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Rewrite the two-children paradox replacing «boy» with «boy born on a Tuesday»: the answer changes. Discuss why the additional information «on a Tuesday» further restricts the sample space.

Solution

Set-up. In the classic paradox, knowing that «at least one of the two children is a boy» brings the probability that both are boys to $1/3$ (not $1/2$), because the sample space of ordered pairs is $\{MM, MF, FM, FF\}$ and the condition removes only $FF$. Adding the day of birth, each child has $2\times 7 = 14$ possible «types» (sex × day), so the sample space has $14\times 14 = 196$ equiprobable ordered pairs. Condition on «at least one child is a boy born on a Tuesday»: count the pairs that satisfy the condition (use inclusion–exclusion so as not to count twice the case in which both are) and, among these, those with two boys. The ratio gives a probability close to $13/27$, appreciably different from $1/3$. The conceptual point: more specific information identifies a smaller and «less symmetric» subset of the sample space, and this shifts the conditional probability.

Links

Topics: Probabilita
Concepts: Paradosso dei due figli · Probabilita condizionata · Spazio campione
Skills: Calcolo probabilita · Ragionare per casi
Exercise type: Problema probabilita