Problem 24 - Algebra and Trigonometry

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 \(\\{M M M, M M F, M F M, M F F, F M M FMF, FFM, FFF\)} - Find the probability of selecting a family with $$\text{at least two female children.}$$

Step-by-Step Solution

Answer

The probability of selecting a family with at least two female children is 0.5 or 50%.

Step 1: Identify total number of outcomes

There are a total of 8 possible combinations for 3 children, which are: MMM, MMF, MFM, MFF, FMM, FMF, FFM, FFF.

Step 2: Identify favorable outcomes

Next, look for combinations that have at least two female children. These combinations are: MFF, FMF, FFM and FFF. So there are 4 favorable outcomes.

Step 3: Calculate Probability

Probability is calculated by dividing the number of favorable outcomes by the total number of outcomes. So, the probability in this case is \(\frac{4}{8} = 0.5\) or 50%.

Key Concepts

Sample Space

The concept of a sample space is fundamental in probability theory. It refers to the set of all possible outcomes for a particular experiment or situation. In this context, we are considering the genders of three children in a family. Each child can either be male (M) or female (F). Hence, when you list out all the combinations of three children, you get a sample space:

• MMM
• MMF
• MFM
• MFF
• FMM
• FMF
• FFM
• FFF

These eight combinations represent all the different possibilities for the genders of the three children, making up the sample space for this problem. Understanding the sample space is crucial as it forms the basis for calculating probabilities.

Favorable Outcomes

Favorable outcomes are those specific outcomes that satisfy the condition we are interested in. In this exercise, we are looking for families with at least two female children.
Within the sample space of eight potential families, we need to identify which outcomes meet this criteria.
Looking at the sample space, the outcomes where there are at least two female children are:

• MFF
• FMF
• FFM
• FFF

There are four such combinations or favorable outcomes in this scenario. Recognizing favorable outcomes helps in the next step, which is calculating the probability.

Equally Likely Outcomes

When dealing with probabilities, it's crucial to assume that all individual outcomes in the sample space are equally likely. This means each possibility has the same chance of occurring. In our problem about the genders of three children, we assume that any given child is equally likely to be male or female.
Thus, each combination of children, whether MMM or FFF, has an equal chance of 1/8, considering there are eight possible outcomes in the sample space.
This assumption of equally likely outcomes allows us to fairly calculate probabilities based solely on counts, as each listing has an unbiased and equal chance.

Probability Calculation

Calculating probability involves determining how likely a favorable outcome is within the sample space. For our example, the probability of a family having at least two female children is found by dividing the number of favorable outcomes by the total number of possible outcomes.To compute this probability, we use the formula:\[P( ext{at least two females}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}\]With 4 favorable outcomes (MFF, FMF, FFM, FFF) and 8 total outcomes, the probability is:\[P( ext{at least two females}) = \frac{4}{8} = 0.5\]This result means there is a 50% chance of selecting a family with at least two female children, perfectly illustrating how probability provides a numeric measure of the likelihood of an event.