Problem 10 - Algebra 2 and Trigonometry

Question

In \(7-14,\) answers can be rounded to four decimal places. A fast-food restaurant gives coupons for 10\(\%\) off of the next purchase with 1 out of every 5 purchases. a. What is the probability that Zoe will receive a coupon with her next purchase? b. What is the probability that Zoe will receive just one coupon with her next three purchases?

Step-by-Step Solution

Answer

a. 0.2 b. 0.3840

Step 1: Understand the Probability of a Coupon

The problem states that Zoe receives a coupon 1 out of every 5 purchases. This implies that the probability of receiving a coupon on a single purchase is \( \frac{1}{5} \) or 0.2.

Step 2: Calculate the Probability of Getting a Coupon

Since we need the probability that Zoe receives a coupon on the next purchase, which is the same as one out of five, we already have it from Step 1 as 0.2.

Step 3: Define the Probability Distribution

Zoe's situation fits a binomial distribution where the probability of success (receiving a coupon) on any purchase is 0.2. We need to find the probability of getting exactly one coupon in three purchases.

Step 4: Calculate Binomial Probability

The formula for finding the probability of exactly \(k\) successes (where \(k = 1\)) in \(n\) trials (where \(n = 3\)) is given by: \[ P(X = k) = \binom{n}{k} \times p^k \times (1-p)^{n-k} \]Substituting in the values, \( p = 0.2 \),\[ P(X = 1) = \binom{3}{1} \times (0.2)^1 \times (0.8)^{2} \]

Step 5: Calculate Combinatorial and Simplify

Calculate the binomial coefficient \( \binom{3}{1} \):\[ \binom{3}{1} = 3 \]Then compute the probability:\[ P(X = 1) = 3 \times 0.2 \times 0.64 \]\[ P(X = 1) = 0.384 \]

Step 6: Round Off the Result

Since the answers should be rounded to four decimal places, the probability that Zoe will receive exactly one coupon in her next three purchases is 0.3840.

Key Concepts

Probability Distribution

A probability distribution describes all the possible outcomes of a random variable and the likelihood of each outcome. In this case, Zoe's chance of receiving a coupon can be modeled using a binomial distribution.

• Binomial distribution is useful when there are only two possible outcomes in each trial—like receiving a coupon or not.
• For Zoe, each purchase is an independent trial with a fixed probability of success, which is 0.2 or 20% for receiving a coupon.

This model helps calculate the probability of different numbers of coupons received in multiple purchases.
Using Zoe's three purchases as trials, you can determine the probability of receiving 0, 1, 2, or 3 coupons using this binomial distribution formula.

Binomial Coefficient

The binomial coefficient, denoted by \( \binom{n}{k} \), is a key component in the binomial probability formula. It represents the number of ways to choose \( k \) successes from \( n \) trials without regard to the order.
For example in Zoe's scenario, we needed to calculate \( \binom{3}{1} \) because we wanted to find the probability of receiving exactly one coupon in three tries. The calculation for \( \binom{3}{1} \) is: \[\binom{3}{1} = \frac{3!}{1!(3-1)!} = 3\]This result tells us there are three different outcomes for Zoe to receive one coupon among three purchases.
Understanding binomial coefficients is crucial for effectively using the binomial probability formula.

Combinatorial Calculations

Combinatorial calculations are at the heart of solving probability problems using the binomial model. These calculations help determine the different combinations in which outcomes can occur. In simpler terms, they allow us to count the number of ways an event can happen.
The use of the formula \( \binom{n}{k} \) is an example of combinatorial calculations and helps us figure out all possible successful arrangements of events. In Zoe’s case:

• We want exactly one coupon, which can occur in different sets of purchases.
• This is where the previously explained binomial coefficient \( \binom{3}{1} = 3 \) comes into play.

The overall calculation gives us a clear picture of how likely specific outcomes are under a given set of conditions.

Rounding Decimals

Rounding decimals is often necessary in probability, especially when precise calculations are required for interpretations. In this exercise, we round our answer to four decimal places to provide an accurate result.
When we calculated the probability that Zoe would receive exactly one coupon in her next three purchases, we found: \[P(X = 1) = 0.384\]However, since the instructions require rounding to four decimal places, we adjust this to: \[P(X = 1) = 0.3840\]Rounding decimals helps us present results in standard formats, ensuring consistency and clarity while interpreting probabilities.