Question

PROBLEM SOLVING At a gas station, \(84 \%\) of customers buy gasoline. Only \(5 \%\) of customers buy gasoline and a beverage. What is the probability that a customer who buys gasoline also buys a beverage?

Short answer

The probability that a customer who buys gasoline also buys a beverage is approximately \( 0.0595 \) or \( 5.95 \% \).

Step-by-step solution

Translate Percentages into Probabilities

Translate the given percentages into probabilities. The probability of a customer buying gasoline, \(P(G)\), is \(0.84\). The probability of a customer buying both gasoline and a beverage, \(P(G \cap B)\), is \(0.05.\)

Calculate Conditional Probability

According to the principle of conditional probabilities, the probability of an event B given that another event A has already occurred can be calculated as: \(P(B|A) = P(A \cap B) / P(A)\). Here, event B is 'Customer buys a beverage' and event A is 'Customer buys gasoline'. Therefore, to find the probability that a customer who buys gasoline also buys a beverage, plug the known values into the equation: \( P(B|G) = P(G \cap B) / P(G) = 0.05 / 0.84\).

Interpret the Result

The result obtained in Step 2 should be translated into a percentage by multiplying it by 100, which gives the probability that a customer who buys gasoline will also buy a beverage.

Key concepts

Understanding Probability and Statistics

Probability and statistics are branches of mathematics that deal with analyzing the likelihood of events and the collection, analysis, interpretation, and presentation of data. In everyday life, we use the concept of probability often, such as determining the chance of rain or winning a game. In statistics, probability is used as a tool to predict and make decisions based on data.

In the given exercise, we consider the event of a customer buying gasoline at a gas station and the event of the same customer also buying a beverage. To calculate the likelihood or probability of these events, we use the percentages given. The language of probability allows us to quantify these chances in a mathematical way. Understanding the basic principles of probability can drastically improve our ability to make informed decisions based on quantitative data.

Percentages in Probability

Percentages in probability provide a way to express the likelihood of an event in terms familiar to most people. Converting a percentage into a probability gives a value between 0 and 1, with 0 indicating impossibility and 1 indicating certainty. For example, a 50% chance corresponds to the probability of 0.5. In our exercise, the percentage of customers buying gasoline (84%) and the percentage buying both gasoline and a beverage (5%) are converted to probabilities (0.84 and 0.05 respectively).

Conversion from Percentage to Probability

The formula to convert a percentage to a probability is simple: divide the percentage by 100. If we say there is an 84% chance of an event occurring, in probability terms, we write this as 0.84. This fundamental concept is crucial as all further calculations in probability, including those involving conditional probability, depend on working with these decimal values rather than percentages.

Conditional Probability Calculation

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. This type of probability can be especially helpful in scenarios where the outcome might be affected by the presence of another related event. In the problem provided, we are asked to find the likelihood that a customer buys a beverage given that they have already bought gasoline.

Formula for Conditional Probability

The calculation for conditional probability, as seen in our step-by-step solution, is defined by the formula: \[\begin{equation} P(B|A) = \frac{P(A \cap B)}{P(A)} \end{equation}\]Here, \(P(B|A)\) is the probability of event B occurring after event A has occurred, \(P(A \cap B)\) is the joint probability of events A and B happening together, and \(P(A)\) is the probability of event A occurring. By inserting the probabilities obtained from the percentages, we can compute the conditional probability, which not only enhances our understanding of the immediate problem but also provides us with a powerful analytical tool for interpreting real-world situations where events are interdependent.