Problem 24 - Algebra and Trigonometry

Question

Education Level In a certain county 20\(\%\) of the population have a college degree. A jury consisting of 12 people is selected at random from this county. (a) What is the probability that exactly two of the jurors have a college degree? (b) What is the probability that three or more of the jurors have a college degree?

Step-by-Step Solution

Answer

(a) 0.2835 (b) 0.4417

Step 1: Identify the Problem Type

This problem involves calculating probabilities related to a binomial distribution. The probability of having a college degree (success) is 20%, or 0.20, and the jury consists of 12 people (trials).

Step 2: Setting Up the Binomial Formula

The binomial probability formula is used, where \( n = 12 \) (number of trials), \( k \) is the number of successes (college degrees), and \( p = 0.2 \) is the probability of a single success. The formula is: \[ P(X = k) = \binom{n}{k} \, p^k \, (1-p)^{n-k} \]

Step 3: Calculate the Probability for Exactly Two College Degrees

Set \( k = 2 \) for exactly two college degrees and substitute into the formula: \[ P(X = 2) = \binom{12}{2} \, (0.2)^2 \, (0.8)^{10} \] Calculate the binomial coefficient: \[ \binom{12}{2} = \frac{12 \times 11}{2} = 66 \] Substitute back to get: \[ P(X = 2) = 66 \, \times \, (0.2)^2 \, \times \, (0.8)^{10} \] Calculate this value to find \( P(X = 2) \approx 0.2835 \).

Step 4: Calculate the Probability for Three or More College Degrees

Here we want \( P(X \geq 3) \). Since directly calculating three or more can be complex, use the complement rule: \[ P(X \geq 3) = 1 - P(X < 3) \] Calculate \( P(X = 0) \) and \( P(X = 1) \) using the formula:\[ P(X = 0) = \binom{12}{0} \, (0.2)^0 \, (0.8)^{12} \approx 0.0687 \]\[ P(X = 1) = \binom{12}{1} \, (0.2)^1 \, (0.8)^{11} \approx 0.2061 \]Find \( P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \approx 0.5583\).Finally, \[ P(X \geq 3) = 1 - 0.5583 \approx 0.4417 \].

Key Concepts

Probability Calculation

When dealing with probabilities, the goal is to determine how likely certain events are to happen. Probability calculation is essential when you're working with random events involving a specific number of trials or attempts. In this context, we aim to evaluate the likelihood of specific outcomes within a set number of occurrences.
For instance, if you're interested in the probability of exactly two jurors out of twelve having a college degree, you first need to understand what each potential outcome signifies. Each juror can either have a degree or not, which makes this a yes-or-no type outcome.
In statistical terms, each juror is considered a trial with the possibility of success (having a degree) or failure (not having a degree). The probability of these outcomes can be calculated using methods best suited for this structure, such as the binomial distribution. Calculating probabilities precisely often involves understanding and applying specific statistical techniques or formulas, like the ones we'll explore next.

Binomial Formula

The binomial distribution is key to tackling problems where you have a fixed number of trials, each with two possible outcomes: success or failure. It's particularly useful when you're dealing with independent events, such as jurors being randomly chosen.
To find the probability of a certain number of successes, like exactly two jurors with a degree, you use the binomial formula: \[ P(X = k) = \binom{n}{k} \times p^k \times (1-p)^{n-k} \] where:

• \( n \) represents the number of trials (the jury size, which is 12).
• \( k \) is the desired number of successes (jurors with a degree, for example, 2).
• \( p \) is the success probability for each trial (0.2 in this problem).

This formula calculates the probability of exactly \( k \) successes in \( n \) independent trials. It combines the probability of a specific configuration of successes occurring along with all possible ways that \( k \) successes could occur out of \( n \) trials. It's powerful because it applies to many practical situations, like probability exercises involving random selections.

Complement Rule

The complement rule is a handy tool in probability when it is easier to calculate the probability of something not happening. Instead of calculating a more complex probability directly, you compute the likelihood of the opposite event and subtract it from 1.
For instance, if you need to find the probability of three or more jurors having a degree, computing this directly involves multiple steps. Instead, by using the complement rule, you find the probabilities of scenarios with fewer than three degrees and subtract that from 1.
This is expressed mathematically as: \[ P(X \geq 3) = 1 - P(X < 3) \] where \( P(X < 3) \) involves adding probabilities of 0, 1, or 2 successes. This method can save time and reduce errors, especially when dealing with large datasets or complicated scenarios. The complement rule thus simplifies the process and ensures precision in probability assessments.