Problem 29 - Algebra 2

Question

CARS According to a recent survey, about 1 in 3 new cars is leased rather than bought. What is the probability that 3 of 7 randomly selected new cars are leased?

Step-by-Step Solution

Answer

The probability is \( \frac{560}{2187} \).

Step 1: Understand the Problem

We need to find the probability that 3 out of 7 randomly selected cars are leased, given that the probability of leasing one car is \( \frac{1}{3} \). This is a binomial probability distribution problem.

Step 2: Set Up the Binomial Formula

The probability of exactly \( k \) successes in \( n \) independent trials is given by the binomial formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \( n = 7 \), \( k = 3 \), and \( p = \frac{1}{3} \).

Step 3: Calculate the Binomial Coefficient

Calculate the binomial coefficient \( \binom{7}{3} \), which represents the number of ways to choose 3 leased cars out of 7. \[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = 35 \]

Step 4: Compute the Probability of Successes

Calculate the term \( p^k \) which is the probability of having 3 leased cars:\[ p^k = \left(\frac{1}{3}\right)^3 = \frac{1}{27} \]

Step 5: Compute the Probability of Failures

Calculate the term \( (1-p)^{n-k} \) which is the probability of the remaining cars not being leased:\[ (1-p)^{n-k} = \left(\frac{2}{3}\right)^{4} = \frac{16}{81} \]

Step 6: Combine Everything to Find the Solution

Substitute the values calculated into the binomial formula:\[ P(X = 3) = 35 \times \frac{1}{27} \times \frac{16}{81} \]\[ P(X = 3) = 35 \times \frac{16}{2187} \]\[ P(X = 3) = \frac{560}{2187} \]

Step 7: Simplify the Fraction

Ensure the fraction \( \frac{560}{2187} \) is in its simplest form. Since both numbers have no common factors other than 1, this fraction is already simplified.

Key Concepts

Binomial Coefficient

In the world of probability and statistics, the binomial coefficient is a fundamental concept. It's used to determine how many different ways we can choose a specific number of successes in a set number of trials. Think of it as figuring out how many different ways you can pick a certain number of apples if you have a basket of apples of a fixed size. For example, suppose you want to choose 3 apples from a basket containing 7 apples. The formula for calculating the binomial coefficient is given by:\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]Here, \( n \) is the total number of items (or trials), and \( k \) is the number of items (or successes) we want to choose. In our exercise, \( n = 7 \) and \( k = 3 \). By plugging these values into the formula, we find that:\[ \binom{7}{3} = \frac{7!}{3! \times 4!} = 35 \]This tells us there are 35 different ways to choose 3 leased cars from 7 new cars.

Probability of Success and Failure

When dealing with a binomial probability distribution, it's essential to understand the terms "probability of success" and "probability of failure." - **Probability of Success**: This represents the likelihood of the event we are interested in occurring. In our problem, this is the probability of a car being leased, which is given as \( \frac{1}{3} \).- **Probability of Failure**: This is the chance of the opposite event happening. For the cars not being leased, the probability of failure is \( 1 - \frac{1}{3} = \frac{2}{3} \).In a binomial setting, you calculate the probability of having exactly \( k \) successes (e.g., 3 leased cars) in \( n \) trials (e.g., choosing 7 cars). The formula used is:\[ p^k = \left(\frac{1}{3}\right)^3 = \frac{1}{27} \]And the probability of failure for the rest of the trials is:\[ (1-p)^{n-k} = \left(\frac{2}{3}\right)^4 = \frac{16}{81} \]This approach helps in calculating how likely it is to end up with a specified number of successes and failures.

Simplifying Fractions

Simplifying fractions is a crucial step in making numerical solutions more approachable and comprehensible. It involves reducing a fraction to its most basic level by finding the greatest common divisor (GCD) of its numerator and denominator.When you simplify a fraction, you keep the relationship of the numbers intact but in a more straightforward form. Take for instance the final fraction from our exercise:\[ \frac{560}{2187} \]Here, both 560 and 2187 do not share any common factors other than 1. So, this fraction is already in its simplest form. Nevertheless, it's important to check this for any fraction to ensure clarity and accuracy in mathematical communication.When presented with a complex numerical fraction, remember to:- Check for common factors- Divide both the numerator and the denominator by the GCDThese steps make calculations neater and can often simplify further computations.