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Chapter 5: Problem 13

About \(13 \%\) of the population is left-handed. If two people are randomly selected, what is the probability that both are left-handed? What is the probability that at least one is right-handed?

Short Answer

Expert verified
The probability that both people are left-handed is 0.0169. The probability that at least one is right-handed is 0.9831.

Step by step solution

01

Determine the Probability of One Person Being Left-Handed

The probability that one person is left-handed is given as 13%, which can be written as a decimal: \( P(\text{Left-Handed}) = 0.13 \).
02

Calculate the Probability that Both People Are Left-Handed

To find the probability that both people are left-handed, multiply the probabilities for each person, assuming the events are independent:\[ P(\text{Both Left-Handed}) = 0.13 \times 0.13 = 0.0169 \].
03

Determine the Probability of One Person Being Right-Handed

The probability that one person is right-handed is the complement of the probability that the person is left-handed:\[ P(\text{Right-Handed}) = 1 - P(\text{Left-Handed}) = 1 - 0.13 = 0.87 \].
04

Calculate the Probability that at Least One Person is Right-Handed

To find the probability that at least one of the two people is right-handed, use the complement rule. First, find the probability that both are left-handed (from Step 2), then subtract this from 1:\[ P(\text{At Least One Right-Handed}) = 1 - P(\text{Both Left-Handed}) = 1 - 0.0169 = 0.9831 \].

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Key Concepts

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

left-handed probability
Left-handedness is less common in the general population. Around 13% of people are left-handed. When we express this probability in decimal form, it becomes 0.13. This can help in various probability calculations. For example, knowing the chance of one person being left-handed is the starting point for more complex scenarios. Understanding basic probabilities like this is crucial for solving problems involving multiple events.
independent events
In probability, independent events mean the outcome of one event does not affect the outcome of another. If we consider two random people, the probability of each being left-handed is independent of the other. To find the probability that both individuals are left-handed, we multiply their individual probabilities: \[ P(\text{Both Left-Handed}) = P(\text{Left-Handed}) \times P(\text{Left-Handed}) = 0.13 \times 0.13 = 0.0169 \] Recognizing events as independent simplifies calculations and is essential in multi-step problems.
complement rule
The complement rule is a crucial concept in probability. It states that the probability of an event not occurring is 1 minus the probability of the event occurring. For example, if 13% of people are left-handed, then 87% are not. Expressed in probability terms:\[ P(\text{Right-Handed}) = 1 - P(\text{Left-Handed}) = 1 - 0.13 = 0.87 \] This rule helps find probabilities indirectly and is useful when calculating 'at least' probabilities, like the chance that at least one person in a group is right-handed.
probability of complementary events
Complementary events are pairs of outcomes where one must occur if the other does not. For instance, someone being left-handed or right-handed are complementary events. To find the probability that at least one of two people is right-handed, we first find the chance both are left-handed (using the independent events concept) and then apply the complement rule:\[ P(\text{At Least One Right-Handed}) = 1 - P(\text{Both Left-Handed}) = 1 - 0.0169 = 0.9831 \] Thus, there's a very high probability (98.31%) that at least one of the two people is right-handed. Understanding these complement relationships simplifies complex probability questions.

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