Answer to Riddle #4: Mythical City Population of Boys

The percentage of children that are male at the end of year t is expected to evolve through time as follows:

• In the first year, the percentage of children that are male will be 50%. This is because there is an equal chance of a child being born male or female.
• In the second year, the percentage of children that are male will be slightly higher than 50%. This is because some of the families that had all girls in the first year will have a boy in the second year.
• In the third year, the percentage of children that are male will be even higher than 50%. This is because even more of the families that had all girls in the first two years will have a boy in the third year.
• This trend will continue, and the percentage of children that are male will gradually increase over time.

However, it is important to note that this is just an expected value. The actual percentage of children that are male could be higher or lower, depending on the random chance of which families have boys and which families have girls.

The below answer IS WRONG
At the start, there are 100,000 couples, which means that there are 100,000 babies born in the first year. Since each baby has an equal chance of being male or female, we expect half of them to be male. Therefore, p(1) = 50%.

Now, let's consider the situation at the end of the second year. Of the 100,000 couples, roughly half will have had a boy and will not have any more children. The other half will have had a girl and will continue having children. Therefore, we can expect around 50,000 families to have a baby in the second year. Once again, each baby has an equal chance of being male or female, so we expect half of the babies to be male. However, we also need to take into account the fact that some families will continue having children until they have a boy. This means that some families that had girls in the first year will continue having children in the second year. The probability that a family that had a girl in the first year will have a boy in the second year is 1/2. Therefore, we can expect around 25,000 families to have a boy in the second year.

Putting this all together, we can see that at the end of the second year, there will be a total of 75,000 children, half of whom are male and half of whom are female. Therefore, p(2) = 50%.

We can repeat this reasoning for subsequent years. At the end of the third year, we expect around 50,000 families to have a baby, and 1/2 of these babies will be male. In addition, we expect around 25,000 families to have a boy in the second year and then stop having children, and we expect around 12,500 families to have a girl in the first two years and then have a boy in the third year. Therefore, we can expect a total of 87,500 children at the end of the third year, of which around 53.57% are male. Therefore, p(3) ˜ 53.57%.

We can continue this reasoning for subsequent years. At the end of the fourth year, we expect a total of 93,750 children, of which around 57.81% are male. At the end of the fifth year, we expect a total of 96,875 children, of which around 59.77% are male. At the end of the sixth year, we expect a total of 98,438 children, of which around 60.74% are male. As time goes on, the percentage of male children is expected to approach 61.8%, which is the limit as t goes to infinity.

In summary, the percentage of male children is expected to start at 50% and gradually increase as time goes on, approaching a limit of 61.8% as t goes to infinity.

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