Black and White Pebbles
We have 100 pebbles, 50 of them are white and 50 of them are black.
We distribute them in two bags so that no bag is empty. We choose one of
the bags uniformly at random and take a pebble from that bag uniformly
at random. What is the minimum and maximum probability that the pebble
is white? Notice that we distributed the pebbles ourselves and we know
the distribution before the random selection.
Link to the problem on Twitter: https://twitter.com/Riazi_Cafe/status/1666794242683858944
The minimum probability is when one bag has only one black pebble and the rest of the pebbles are in the other bag. In this case the probability of choosing a white pebble is \(50/198 \simeq 0.2525\).
The maximum probability is when one of the bags has one white pebble
and the rest of the pebbles are in the other bag. In this case the
probability of choosing a white pebble is \(1/2 + 49/198 \simeq 0.7474\).
Link to the solution on Twitter: https://twitter.com/Riazi_Cafe/status/1667340991710523392