Problem 22
Question
You select a family with three children. If \(M\) represents a male child and \(F\) a female child, the sample space of equally likely outcomes is \(\\{M M M, M M F, M F M\) \(M F F, F M M, F M F, F F M, F F F\\} .\) Find the probability of selecting a family with: at least two female children.
Step-by-Step Solution
Verified Answer
The probability of selecting a family with at least two female children is 0.5
1Step 1: Identify the Total Number of Outcomes
According to the problem, the total number of outcomes in the sample space is 8: \(\{M M M, M M F, M F M, M F F, F M M, F M F, F F M, F F F\}\)
2Step 2: Identify the Favorable Outcomes
We are to find the families with 'at least two female children'. The favorable outcomes in this case are: \(\{M F F, F M F, F F M, F F F\}\). These 4 outcomes contain at least two female children.
3Step 3: Calculate the Probability
Probability is calculated as the ratio of favorable outcomes to total outcomes. The total outcomes is 8 and the number of favorable outcomes is 4. Therefore, the probability of selecting a family with at least two female children is \(\frac{4}{8} = 0.5\)
Key Concepts
Sample SpaceFavorable OutcomesOutcomes in Probability
Sample Space
Understanding the concept of a sample space is fundamental when studying probability. A sample space is the set of all possible outcomes in a probability experiment. For example, when considering a family with three children, each child can either be male (denoted as \(M\)) or female (denoted as \(F\)). Hence, every combination of three children forms distinct outcomes.
In the original exercise, the sample space consists of eight possible combinations of these children:
In the original exercise, the sample space consists of eight possible combinations of these children:
- MMM - all three children are male
- MMF - two males, one female
- MFM - male, female, male
- MFF - male and two females
- FMM - female and two males
- FMF - female, male, female
- FFM - two females, one male
- FFF - all three children are female
Favorable Outcomes
Once you have established the complete sample space, the next step in solving probability problems is identifying favorable outcomes. Favorable outcomes are those specific results in the sample space that satisfy the conditions of the probability question you are trying to answer. In this context, they are the outcomes in which the condition 'at least two female children' is met.
From the sample space provided, favorable outcomes for this condition are:
From the sample space provided, favorable outcomes for this condition are:
- MFF - one male, two females
- FMF - female, male, female
- FFM - two females, one male
- FFF - all three children are female
Outcomes in Probability
In probability theory, outcomes are the individual results that comprise the sample space. Each outcome is considered equally likely unless specified otherwise. Thus, the focus lies on determining the likelihood of an event based on these outcomes. Probability is expressed as a fraction, the ratio of favorable outcomes to total possible outcomes.
The original exercise guides us through calculating the probability by first identifying both the number of favorable outcomes and the total number of outcomes. Here, the total number of outcomes is 8, as established by the complete sample space. The favorable outcomes number 4. Therefore, the probability of choosing a family with at least two female children is constructed through the formula:\[P(\text{at least 2 female}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} = \frac{4}{8} = 0.5\]Equating to a probability of 0.5 or 50%, this approach provides insight into how likely it is for at least two of the three children to be female. Properly identifying and counting these outcomes dictates the accuracy of your probability calculations.
The original exercise guides us through calculating the probability by first identifying both the number of favorable outcomes and the total number of outcomes. Here, the total number of outcomes is 8, as established by the complete sample space. The favorable outcomes number 4. Therefore, the probability of choosing a family with at least two female children is constructed through the formula:\[P(\text{at least 2 female}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} = \frac{4}{8} = 0.5\]Equating to a probability of 0.5 or 50%, this approach provides insight into how likely it is for at least two of the three children to be female. Properly identifying and counting these outcomes dictates the accuracy of your probability calculations.
Other exercises in this chapter
Problem 21
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