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Chapter 11: Problem 29

You select a family with three children. If \(M\) represents a male child and \(F\) a female child, the set of equally likely outcomes for the children's genders is \\{MMM, MMF, MFM, \(M F F, F M M, F M F, F F M, F F F\\}\). Find the probability of selecting a family with exactly two male children.

Short Answer

Expert verified
The probability of selecting a family with exactly two male children is \(\frac{3}{8}\).

Step by step solution

01

Identify Possible Outcomes

Firstly, list down all possible outcomes when a family has three children. According to the problem these are \[\{MMM, MMF, MFM, MFF, FMM, FMF, FFM, FFF\}\]
02

Identify Favorable Outcomes

Next, identify the outcomes where the family has exactly two male children. These are \[\{MMF, MFM, FMM\}\]
03

Compute Probability

Calculate the probability by dividing the number of favorable outcomes by the number of total possible outcomes. The total possible outcomes are 8 and favorable outcomes are 3, so the required probability is \(\frac{3}{8}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics
Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. It helps us determine how many ways things can happen. In our exercise, we need to enumerate possible outcomes when considering different cases, like gender combinations in a family of three children.
  • First, we list all possible outcomes of the genders: - Male Male Male (MMM) - Male Male Female (MMF) - Male Female Male (MFM) - Male Female Female (MFF) - Female Male Male (FMM) - Female Male Female (FMF) - Female Female Male (FFM) - Female Female Female (FFF)
Each sequence is unique, and all possible combinations need to be considered. Combinatorics simplifies this process by providing strategies to account for every possibility. With practice, combinatorics becomes easier and helps solve complex probability problems efficiently.
Gender selection
Gender selection in probability exercises refers to determining the likelihood of specific gender combinations in a family or group. It involves identifying all possible gender configuration outcomes. In our exercise, we focused on a family with three children and wanted a group with exactly two male children.
  • We need to count the variations where there are two males: - Male Male Female (MMF) - Male Female Male (MFM) - Female Male Male (FMM)
These are the 'favorable' outcomes for the problem. When conducting gender selection calculations, one should pay close attention to the gender combination criteria. The ability to discern these criteria quickly is key to mastering probability problems.
Mathematical outcomes
Mathematical outcomes in probability are different possibilities that can occur in a given scenario. Understanding these outcomes is crucial in finding solutions to probability problems. Here, we calculated the chances of picking the family configuration with two male children from the total set of possible gender configurations.
  • We identified 8 total possible outcomes (as listed before).
  • We identified 3 favorable outcomes (only those with two male children).
  • We then calculated the probability of our desired outcome: - Probability = (Number of Favorable Outcomes) / (Total Number of Outcomes) - Probability = \(\frac{3}{8}\)
This calculation summarizes how mathematical outcomes effectively lead to the probability of events. By understanding total versus favorable outcomes, probability becomes a clear, intuitive process.

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