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Chapter 11: Problem 34

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 fewer than four female children.

Short Answer

Expert verified
The probability of selecting a family with less than four female children is \(\frac{7}{8}\)

Step by step solution

01

Identify the total number of outcomes.

There are 8 possible outcomes as given in the problem: MMM, MMF, MFM, \(MFF, FMM, FMF, FFM, FFF\)
02

Identify the favourable outcomes.

The event defined in this exercise (fewer than four female children) includes all possibilities except for FFF - because FFF is the only case with four female children. So the favourable outcomes are: MMM, MMF, MFM, \(MFF, FMM, FMF, FFM\)
03

Calculate the probability.

The probability is calculated by ratio of the number of favourable outcomes to total outcomes. In this case, the total number of outcomes is 8 and the number of favourable outcomes is 7. Hence the probability is \(\frac{7}{8}\)

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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 studies ways of counting, ordering, and arranging objects. It's fundamental to probability theory, where it helps determine the number of possible outcomes of an event.
In this exercise, we use combinatorics to calculate all possible combinations of gender outcomes for three children in a family. Here, each child can either be male (M) or female (F). By considering each child independently, we find the total outcomes as a product of choices.
For three children, there are two choices per child, leading to a total of \(2 \times 2 \times 2 = 8\) possible gender combinations.
Sample Space
The sample space is the set of all possible outcomes of an event. In probability, it's crucial to identify this space to understand how likely different outcomes are.
For our family with three children, the sample space consists of the eight gender combinations: MMM, MMF, MFM, MFF, FMM, FMF, FFM, FFF. This collection represents every possible way the children can be male or female.
By defining this comprehensive list, we ensure that we consider all possibilities when calculating probabilities.
Gender Probability
Gender probability in this context refers to the likelihood of various gender combinations among the children. We look at the probability of specific events related to gender, such as having fewer than four female children.
Given that each child has a 50% chance of being male or female, and since these are independent events, all gender combinations created from the choices have equal probability. Thus, each outcome in our sample space has a probability of \(\frac{1}{8}\).
To find the probability of having fewer than four female children, we use favourable outcomes, excluding the FFF case, resulting in a probability of \(\frac{7}{8}\).
Counting Outcomes
Counting outcomes involves systematically determining every possible result of an experiment. It's tied to the principles of combinatorics, enabling precise probability calculations.
In our scenario, we listed all gender combinations by changing each child's gender while ensuring all options are represented. There were no duplicate representations, which would skew probabilities.
  • MMM - all male
  • MMF, MFM, FMM - two male, one female
  • MFF, FMF, FFM - one male, two female
  • FFF - all female
By correctly counting and organizing these outcomes, we accurately reflect the true scope of possibilities, helping us compute probability effectively.

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Most popular questions from this chapter

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