Chance of a Girl Having a Sister

Solution & Explanation

To solve the problem of determining the probability that a random girl has at least one sister, we need to consider the distribution of family sizes and the probability that a girl from each family size has a sister.

Step-by-Step Solution:

1. Understanding the Probability Distribution:

   The table given provides the probability distribution of the number of children per family:

   n_children	0	1	2	3	4	>=5
   p	0.3	0.25	0.2	0.15	0.1	0

   • This means 30% of families have no children, 25% have 1 child, 20% have 2 children, 15% have 3 children, and 10% have 4 children. Families with 5 or more children are not considered.

2. Probability of Having a Sister:

   For a girl to have a sister, there must be at least 2 children in the family. Therefore, we will consider families with 2 or more children.

   • Families with 2 children:

     • Possible combinations: GG, GB, BG, BB
     • Probability of having at least one sister (GG): 1/2
     • Contribution to overall probability: 0.2 * 1/2 = 0.1

   • Families with 3 children:

     • Possible combinations: GGG, GGB, GBG, BGG, GBB, BGB, BBG, BBB
     • Probability of having at least one sister: 1 - Probability of all siblings being boys = 1 - (1/2)^2 = 3/4
     • Contribution to overall probability: 0.15 * 3/4 = 0.1125

   • Families with 4 children:

     • Probability of having at least one sister: 1 - Probability of all siblings being boys = 1 - (1/2)^3 = 7/8
     • Contribution to overall probability: 0.1 * 7/8 = 0.0875

3. Combining the Probabilities:

   We sum the contributions from families with 2, 3, and 4 children to find the total probability that a girl has at least one sister:

   $P(S) = 0.1 + 0.1125 + 0.0875 = 0.3$

   Therefore, the probability that a randomly chosen girl has at least one sister is 30%.

Explanation:

• Why Ignore Families with 0 or 1 Child?

  • A family with 0 children doesn't exist in the context of the question.
  • A family with 1 child means the girl has no siblings, thus no sister.

• Why Consider Only 2 or More Children?

  • Only families with 2 or more children can have more than one girl, hence the possibility of having a sister.

• Probability of Having a Sister:

  • For each family size, the probability of having a sister is calculated by considering the possible gender combinations and excluding those with no sisters. This is weighted by the probability of selecting a family of that size.

This approach ensures a comprehensive solution by considering all possible scenarios where a girl can have a sister, weighted by the likelihood of each family size.