There are 300 balls, 150 of which are white and the remaining of them are black. These balls are mixed up and put in three separate boxes such that each box contains 100 of these balls. However, we do...

There are 300 balls, 150 of which are white and the remaining of them are black. These balls are mixed up and put in three separate boxes such that each box contains 100 of these balls. However, we do not know how many white balls there are in each box. Then, we toss a fair die and:

• if the upturned face is 1, we randomly choose a ball from the first box,


• if the upturned face is 2 or 3, we randomly choose a ball from the second box,


• if the upturned face is 4, 5, or 6, we randomly choose a ball from the third box.

In the end of the procedure, it is known that a white ball is selected. A friend, who knows the color distribution in each box, has told us:

• given that you select a white ball, the probability that the die output was a 1 is19" role="presentation" style="display: inline; line-height: normal; font-size: 14.300000190734863px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">1919,


• given that you select a white ball, the probability that the die output was a 2 is1363" role="presentation" style="display: inline; line-height: normal; font-size: 14.300000190734863px; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">13631363,