Puzzle | Tuesday Boy Paradox

Ankur’s elder brother got married 10 years ago. This summer, Ankur is going to meet his brother after a decade. As a token of love, he plans to buy clothes for his brother’s kids.

From what is remembered, Ankur's brother has two children, and it is known that:

• One of the children is a boy, and that boy was born on a Tuesday.

Assume:

1. Each child is equally likely to be a boy (B) or a girl (G).
2. Each child is equally likely to be born on any of the 7 days of the week, independent of sex.

What is the probability that both of Ankur’s brother’s children are boys, given that at least one of them is a boy born on a Tuesday?

Check if you were right - full answer with solution below.

Solution:

Possible Outcomes: Without considering the day of birth, there are four equally likely possibilities for the sexes of Ankur’s brother's two children:

• Both are boys (BB).
• One boy (older) and one girl (BG).
• One girl (older) and one boy (GB).
• Both are girls (GG).

However, since we know that at least one of the children is a boy born on a Tuesday, we eliminate the (GG) case because there must be at least one boy.

We are now left with three cases:

• BB (both boys),
• BG (older boy, younger girl),
• GB (older girl, younger boy).

Considering Days of the Week:

Each child could have been born on any of the seven days of the week.
Thus, for each child, there are 7 possible days, which means for two children, there are 7 × 7 = 49 possible birth day combinations for each of the three remaining cases (BB, BG, GB).

Condition: At Least One Boy Born on Tuesday:

For the BB case (both children are boys), there are two ways that at least one of them is born on Tuesday:

• The first boy is born on Tuesday, and the second boy is born on any of the 7 days (7 possibilities).
• The second boy is born on Tuesday, and the first boy is born on any of the 7 days (7 possibilities).

However, if both boys are born on Tuesday, we've double-counted this scenario. So, we subtract 1 from the total, leaving us with 7 + 7 − 1 = 13 favorable outcomes for the BB case.

For the BG case (one boy, one girl), only the boy could be born on Tuesday, so there are 7 possibilities for the boy (born on Tuesday) and 7 possibilities for the girl (born on any day), giving 7 × 7 = 49 possible outcomes. However, since we know that the boy must be born on Tuesday, there are exactly 7 favorable outcomes for the BG case.

For the GB case (one girl, one boy), we only care about the boy being born on Tuesday. As in the BG case, there are exactly 7 favorable outcomes for this case.

Calculating the Total Number of Favorable Outcomes:

The total number of favorable outcomes (cases where there is at least one boy born on Tuesday) is (7 + 7 - 1 = 13) 13 from the BB case, 7 from the BG case, and 7 from the GB case. Thus, the total number of favorable outcomes is 13 + 7 + 7 = 27.

Calculating the Desired Probability: The probability that both children are boys, given that at least one of them is a boy born on Tuesday, is the ratio of the number of favorable outcomes for the BB case (13) to the total number of favorable outcomes (27):

Probability = 13/27

The probability that Ankur’s elder brother has two boys, given that one of them is a boy born on Tuesday, is 13/27, which is a little less than 1/2. This unexpected result is derived from the subtle interactions between the conditions of the problem, particularly involving the days of the week.