4. Probability
Basic Concepts of Probability
2:05 minutesProblem 4.1.29
Textbook Question
In Exercises 29–32, use the given sample space or construct the required sample space to find the indicated probability.
Three Children Use this sample space listing the eight simple events that are possible when a couple has three children (as in Example 2): {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}. Assume that boys and girls are equally likely, so that the eight simple events are equally likely. Find the probability that when a couple has three children, there is exactly one girl.
Verified step by step guidance1
Step 1: Understand the problem. The sample space provided lists all possible outcomes when a couple has three children: {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}. Each outcome represents the gender of the children, where 'b' stands for boy and 'g' stands for girl. We are tasked with finding the probability of having exactly one girl.
Step 2: Identify the outcomes in the sample space that meet the condition of having exactly one girl. To do this, look for outcomes where there is one 'g' (girl) and two 'b's (boys). These outcomes are: {bbg, bgb, gbb}.
Step 3: Count the total number of outcomes in the sample space. The sample space contains 8 outcomes: {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}.
Step 4: Count the number of favorable outcomes. From Step 2, we identified 3 outcomes that satisfy the condition of having exactly one girl: {bbg, bgb, gbb}.
Step 5: Calculate the probability. Since all outcomes are equally likely, the probability of an event is given by the formula: P(Event) = (Number of favorable outcomes) / (Total number of outcomes). Substitute the values from Steps 3 and 4 into this formula to find the probability.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sample Space
A sample space is the set of all possible outcomes of a random experiment. In this context, the sample space consists of all combinations of boys (b) and girls (g) when a couple has three children. For this scenario, the sample space is represented as {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}, which includes eight distinct outcomes.
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Equally Likely Outcomes
Equally likely outcomes refer to situations where each outcome in a sample space has the same probability of occurring. In this problem, since boys and girls are assumed to be equally likely, each of the eight outcomes in the sample space has a probability of 1/8. This uniformity simplifies the calculation of probabilities for specific events.
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Probability of an Event
The probability of an event is calculated as the number of favorable outcomes divided by the total number of possible outcomes in the sample space. To find the probability of having exactly one girl among three children, we identify the favorable outcomes (bbg, bgb, gbb) and divide this by the total outcomes (8), resulting in a probability of 3/8.
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