Problem 20 - Algebra and Trigonometry

Question

Handedness A psychologist needs 12 left-handed subjects for an experiment, and she interviews 15 potential subjects. About 10\(\%\) of the population is left- handed. (a) What is the probability that exactly 12 of the potential subjects are left-handed? (b) What is the probability that 12 or more are left-handed?

Step-by-Step Solution

Answer

(a) Probability is approximately 0.0000009. (b) Probability is approximately 0.000000926.

Step 1: Define the Distribution

The situation can be modeled using a binomial distribution. The number of potential subjects is 15 and the probability of a subject being left-handed is 10%, or 0.1. Let's define X as the number of left-handed subjects among the 15 potential subjects. Here, X follows a Binomial distribution: \( X \sim \text{Binomial}(n=15, p=0.1) \).

Step 2: Calculate Probability of Exactly 12 Left-Handed Subjects

We need to find \( P(X = 12) \). Using the binomial probability formula, \[ P(X = k) = \binom{n}{k} p^k(1-p)^{n-k} \], where \( n = 15 \), \( k = 12 \), and \( p = 0.1 \). Calculating this value: \( P(X = 12) = \binom{15}{12}(0.1)^{12}(0.9)^{3} \). Solving this gives a probability of approximately \( 0.0000009 \).

Step 3: Calculate Probability of 12 or More Left-Handed Subjects

To find \( P(X \geq 12) \), we sum the probabilities of having 12, 13, 14, and 15 left-handed subjects. Using \( P(X = k) = \binom{15}{k} (0.1)^k (0.9)^{15-k} \) again, calculate \( P(X = 13) \), \( P(X = 14) \), and \( P(X = 15) \), then sum them with \( P(X = 12) \). \( P(X = 13) \approx 0.000000026 \), \( P(X = 14) \approx 0.0000000003 \), \( P(X = 15) \approx 0.000000000001 \). Thus, \( P(X \geq 12) \approx 0.0000009 + 0.000000026 + 0.0000000003 + 0.000000000001 \approx 0.000000926 \).

Key Concepts

Probability

Probability is a measure of the likelihood that a particular event will occur. It is calculated as a number between 0 and 1. The closer a probability is to 1, the more likely the event is to happen.

In our scenario, probability helps us determine how likely it is for the psychologist to find a specific number of left-handed individuals in her group of 15 potential subjects.

• A probability of 0 means the event will not happen.
• A probability of 1 means the event is certain to happen.
• A probability of 0.5 indicates that the event is equally likely to happen as it is not to happen.

Understanding probability allows us to make predictions and informed decisions in everyday life.

Binomial Probability Formula

The binomial probability formula is a key tool in statistics used to determine the probability of achieving a specified number of "successes" in a fixed number of independent trials. Here, we have 15 trials (the interviews), and each trial has a 10% chance of being a success (finding a left-handed person).

The formula is: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where:

• \( n \) is the total number of trials.
• \( k \) is the number of successes we're interested in.
• \( p \) is the probability of success on an individual trial.
• \( \binom{n}{k} \) is the binomial coefficient, representing the number of ways to choose \( k \) successes from \( n \) trials.

This formula helps us calculate the exact probability of any particular scenario in binomial contexts, like finding exactly 12 left-handed subjects.

Left-Handedness in Population

Left-handedness is a trait found in approximately 10% of the global population. This is a consistent statistic that varies only slightly across different regions and demographics.

In our exercise, the psychologist's challenge is linked to this percentage, as she seeks to recruit left-handed individuals for her study. Knowing the prevalence of left-handedness helps in making realistic predictions about the potential candidates she might encounter.

• It's interesting to note that this 10% statistic is a result of complex genetic and environmental factors.
• Since left-handedness is less common, statisticians use these kinds of challenges to demonstrate probability and binomial distribution models.

This statistic is crucial, as it provides the base probability for our calculations in the given problem.

Statistical Modeling

Statistical modeling involves using mathematical frameworks to represent complex processes with data. It's a crucial technique in data analysis and decision making.

In our problem, statistical modeling is used to simulate the real-world scenario of finding left-handed subjects through the binomial distribution model.

• This kind of modeling helps us estimate the chances of specific outcomes, like interviewing 12 or more left-handed people.
• It provides insights into the variability and distribution of outcomes over multiple trials.
• Statistical models guide us in understanding patterns and making data-driven predictions.

Overall, statistical modeling allows researchers, such as the psychologist in our problem, to analyze and make informed decisions based on the probabilities of events they want to study.