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Chapter 6: Problem 90

A soft-drink machine dispenses only regular Coke and Diet Coke. Sixty percent of all purchases from this machine are diet drinks. The machine currently has 10 cans of each type. If 15 customers want to purchase drinks before the machine is restocked, what is the probability that each of the 15 is able to purchase the type of drink desired?

Short Answer

Expert verified
The number of Diet Coke customers is \( Diet~Coke~customers = 0.6 \times 15 = 9 \). Since there are 10 cans of Diet Coke, there are enough Diet Coke cans. The number of regular Coke customers is \( Regular~Coke~customers = 15 - 9 = 6 \). There are also 10 cans of regular Coke, so there are enough regular Coke cans. Therefore, the probability that all 15 customers can purchase the drink they want is 100%, or 1.

Step by step solution

01

Calculate the number of customers who want Diet Coke

As 60% of purchases are Diet Coke, and there are 15 customers, we can assume that 60% of these customers will want Diet Coke. Calculate the number of Diet Coke consumers: \( Diet~Coke~customers = 0.6 \times 15 \)
02

Check if there will be enough Diet Coke

We know there are currently 10 cans of Diet Coke in the machine. If there are enough Diet Coke cans for all the customers who want it, each of the 15 customers will be able to purchase their desired type. Check if there will be enough Diet Coke cans: \( Diet~Coke~cans - Diet~Coke~customers \geq 0 \)
03

Calculate the number of customers who want regular Coke

There are 15 total customers, and some want Diet Coke. Calculate the number of regular Coke consumers: \( Regular~Coke~customers = Total~customers - Diet~Coke~customers \)
04

Check if there will be enough regular Coke

We know there are currently 10 cans of regular Coke in the machine. If there are enough regular Coke cans for all the customers who want it, each of the 15 customers will be able to purchase their desired type. Check if there will be enough regular Coke cans: \( Regular~Coke~cans - Regular~Coke~customers \geq 0 \)
05

Determine the probability that all customers can purchase desired drinks

If there are enough cans of each coke type to satisfy the customers wanting that type, then the probability that all 15 customers can purchase the drink they want is 100%, i.e., 1. If there are not enough cans of either coke type, the probability will be 0.

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Key Concepts

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

Conditional Probability
Conditional Probability is a key concept in probability theory. It refers to the probability of an event happening, given that another event has already occurred. In our exercise, we are interested in finding the probability that each customer can get their desired drink, considering the constraints of the drink machine.
When calculating conditional probabilities, it's essential to recognize how existing outcomes affect future ones. For example, in the problem, if a certain number of customers buy Diet Coke, the number of Diet Coke cans decreases, affecting future sales to customers who also want Diet Coke.
Understanding this concept helps in analyzing situations where outcomes depend on particular conditions being met. Conditional Probability can be calculated using the formula:
  • Given events A and B, \[ P(B|A) = \frac{P(A \cap B)}{P(A)} \] This formula helps determine the likelihood of event B occurring if event A has occurred.
Probability Calculation
In probability calculation, we are concerned with determining the likelihood of various events. This involves identifying all possible outcomes and finding the ratio of desired outcomes to total outcomes.
In the exercise, you calculate probabilities through simple arithmetic. Start by knowing how many customers will likely want Diet Coke, using the 60% probability given in the problem. The equation is:
  • \[ Diet\,Coke\,customers = 0.6 \times 15 = 9 \] This tells us 9 customers likely desire Diet Coke.
Next, you verify the quantity of drinks available compared to demand. If each customer's preference can be met with the current stock, the probability is 1 (100%).
Such calculations form the backbone of probability theory. They help assess the likelihood of outcomes both in simple cases, like our exercise, and in more complex statistical scenarios.
Statistics Problem Solving
Statistics problem solving often involves breaking down complex situations into manageable parts to find a solution. Here's a friendly approach to tackle such problems:
First, you determine the given data, such as the proportion of customers preferring Diet Coke. Then, evaluate current resource constraints, such as the availability of the drink. This process requires critical thinking and logical reasoning to piece together problem components step by step. Problem-solving in statistics emphasizes practical application of theories like probability. By methodically assessing each part of the problem, you ensure that all factors are considered. For instance, you:
  • Identify total demand for each drink type.
  • Compare demand against supply to check availability.
  • Conclude if the desired outcome is possible, applying logic and reason.
In our exercise:
  • Calculate demand with existing conditions (9 Diet Coke customers).
  • Check if 10 Diet Coke supplies meet demand.
  • Analyze these steps similarly for regular Coke.
By using these strategies, statistics problem-solving becomes a systematic and structured approach, applicable to various real-world scenarios.

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Most popular questions from this chapter

Suppose that \(5 \%\) of cereal boxes contain a prize and the other \(95 \%\) contain the message, "Sorry, try again." Consider the random variable \(x,\) where \(x=\) number of boxes purchased until a prize is found. a. What is the probability that at most two boxes must be purchased? b. What is the probability that exactly four boxes must be purchased? c. What is the probability that more than four boxes must be purchased?

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