Problem 11 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is: \(\quad\) [April 10, 2019 (I)] (a) \(\frac{1}{11}\) (b) \(\frac{1}{10}\) (c) \(\frac{1}{12}\) (d) \(\frac{1}{17}\)

Step-by-Step Solution

Answer

The answer is \(\frac{1}{11}\), option (a).

Step 1: Understand the Problem

We have two families, each with two children. We need to find out the conditional probability that all children are girls given that at least two are girls in total.

Step 2: List the Possible Outcomes for Each Family

Each family can have children in the combinations: BB (Boy-Boy), BG (Boy-Girl), GB (Girl-Boy), GG (Girl-Girl). Since there are two families, we need to consider combinations for each family.

Step 3: Determine the Sample Space

For two families, each with four possible combinations, the sample space for the two families combined is 16 possible combinations: (BB, BB), (BB, BG), (BB, GB), (BB, GG), (BG, BB), etc.

Step 4: Identify Outcomes Where At Least Two Are Girls

For at least two girls: Cases where either family has GG or one has a girl and the other has at least one girl. The relevant outcomes are: (GG, BB), (BB, GG), (GG, BG), (BG, GG), (GG, GB), (GB, GG), and (GG, GG), (GB, GB), (GB, GG), (GG, GB).

Step 5: Identify Favorable Outcomes

Both families must be GG to have all children as girls. The favorable outcome is only one combination: (GG, GG).

Step 6: Calculate the Conditional Probability

The number of favorable outcomes (GG, GG) is 1. There are 11 outcomes where at least two children are girls. Thus, the conditional probability is \(\frac{1}{11}\).

Step 7: Conclusion

Based on the calculation, the probability that all children are girls given that at least two are girls is \(\frac{1}{11}\).

Key Concepts

Probability Theory

Probability theory is the branch of mathematics that deals with understanding and calculating the likelihood of different outcomes. It helps us manage uncertainty in various situations. In the context of the exercise given, probability theory was used to figure out the chance that all children in two families are girls, given the condition that at least two are girls.

The basic idea in probability is to compare the number of ways an event can happen to the total number of possible outcomes. This is expressed as the formula for probability:

• Probability = \( \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \)

Here, probability theory helped break down the scenario into manageable parts. We calculated the probability by first listing all the possible ways the children could be boys or girls. Then we counted how many of those ways made it possible for at least two children to be girls.

Discrete Mathematics

Discrete mathematics involves studying countable, distinct elements and is used to solve problems on sets, graphs, and probabilities. In this exercise, we utilized discrete mathematics by examining different possible combinations of children's genders in two separate families, which is a finite set of outcomes.

Discrete mathematics is particularly useful when dealing with problems like these where outcomes are distinctly separate. For example, the combinations listed in our solution (like "BB", "BG", "GB", "GG") illustrate discrete possible states each family can be in. Such distinct state counting is essential in determining the sample space and thus the probability in questions like these.

• Combinations of outcomes can be thought of as ordered sets in discrete math.
• Each combination represents a unique element in the sample space.

This approach allowed us to precisely count the possible favorable outcomes and total combinations, helping compute the desired conditional probability.

Combinatorics

Combinatorics is a key concept in probability and discrete mathematics, focusing on counting and arranging objects. It especially comes into play when dealing with problems where you have to choose or arrange different elements. In this problem, combinatorics was used to list all potential gender combinations of children for two families.

Each family having two children results in four possible gender outcomes: Boy-Boy (BB), Boy-Girl (BG), Girl-Boy (GB), and Girl-Girl (GG). By using combinatorial methods, we can systematically arrange these possibilities into a larger set of 16 combinations when considering both families together.

• This count allows for a comprehensive sample space.
• Using combinations, we identified those outcomes where at least two are girls.
• Finally, we used these permutations to determine how many meet the conditions.

Combinatorics is powerful because it provides a structured approach to dissect complex probability scenarios into simpler parts that can be counted or arranged, revealing insights into the likelihood of various events.