Question

For a sales promotion, the manufacturer places winning symbols under the caps of \(10 \%\) of all Pepsi bottles. You buy a six-pack. What is the probability that you win something?

Short answer

The probability of winning something with a six-pack is approximately 46.86%.

Step-by-step solution

Identify the Probability of Winning a Single Bottle

The probability of finding a winning symbol under the cap of each bottle is given as \(10\%\), which in decimal form is \(0.10\).

Calculate the Probability of Not Winning a Single Bottle

The probability of not winning on a single bottle is \(1 - 0.10 = 0.90\). This means there is a 90% chance that any given bottle does not have a winning symbol.

Calculate the Probability of Not Winning in a Six-Pack

Since a six-pack contains 6 bottles and each bottle's outcome is independent, we use the formula for the probability of all independent events occurring: \( (0.90)^6 \). Calculating this gives us \(0.90^6 \approx 0.531441\).

Calculate the Probability of Winning at Least One Bottle

The probability of winning at least one bottle is the complement of not winning any bottle. Thus, it is \(1 - 0.531441 = 0.468559\). Therefore, there is a 46.86% chance of winning something if you buy a six-pack.

Key concepts

Independent Events

In probability, independent events are those whose outcomes do not affect each other. In the context of the given exercise, each Pepsi bottle cap has an independent chance of being a winning cap. This means that the outcome of one bottle does not influence the outcome of any other bottle in the six-pack.

To understand independent events comprehensively, consider rolling a fair die. The result of each roll (e.g., getting a 2 or a 5) does not impact what you might roll next. Similarly, with Pepsi bottles in this exercise, the chance of winning with each bottle remains consistently at 10% regardless of the number of bottles you open or whether previous bottles were winners.

In problems involving independent events like this one, formulas often leverage the independence to simplify calculations, as we'll see in further sections.

Complement Rule

The Complement Rule in probability is a helpful tool for finding the likelihood of at least one or more outcomes occurring by looking at the opposite, or complement, of those outcomes. In simple terms, if you want to determine how likely it is that at least one event will happen, you can find out how likely none will happen and subtract that probability from 1.

For the exercise at hand, to find the probability of winning at least one prize from the six-pack (which consists of 6 Pepsi bottles), we use the complement rule. We first determine the probability that none of the bottles have a winning cap. Using the complement rule, the probability of not winning in any of the bottles is calculated and then subtracted from one.

Understanding complements helps make complicated probability questions much more manageable by breaking them down into simpler steps.

Decimal Probability

Probability can be expressed in various forms: fractions, percentages, and decimals. Decimal probability is particularly convenient for calculation purposes, especially when using computational tools. In this exercise, the probability of finding a winning cap under each bottle has been provided as 10%, which is equivalent to a decimal of 0.10.

Working with decimals allows for straightforward arithmetic, as seen in multiplying probabilities for independent events or complement rule calculations.

For example, if the chance of not winning with a single bottle is 0.90, expressed as a decimal, it is easier to calculate the likelihood of all 6 bottles in a six-pack being non-winners by computing (0.90)^6. This showcases the advantage of using decimals for quick calculations and easy interpretation.

Binomial Probability

The binomial probability specifically addresses scenarios where there are multiple trials of a binary event, meaning each event can result in one of two outcomes.
In a binomial context, questions often ask for the probability of a certain number of successes in a number of trials. Here, finding a winning cap is treated as a 'success', and each bottle cap unopened is a separate trial.

While the solution here directly calculated the complement probability, tackling this as a binomial probability situation can provide deeper insight. We can use the binomial probability formula, where n represents the number of trials (6 bottles), p is the probability of a success (0.10 for each bottle), and k would be the number of successes we are interested in.

For those looking into more complex analysis, the binomial probability model is useful for calculating probabilities across various numbers of successes, thereby extending beyond just 'at least one' success scenarios.

Probability Calculations

Probability calculations form the core of successfully solving these types of problems. Such calculations involve applying rules and formulas effectively. In the exercise, we aimed to find the likelihood of winning at least one bottle in a six-pack of Pepsi through sequences of probability-based steps.

Key calculations included:

• Converting percentage to decimal probability; from 10% to 0.10 for each bottle.
• Using the complement rule to find the probability of no winning caps across all trials.
• Exponentiating the probability of non-success for each independent event [(0.90)^6] to find the chance of losing all rounds.
• Finally, finding the complement or probability of winning at least once by subtracting the all-loss probability from 1 (1−0.531441).

Each step harnesses different probability basics to reach a grand conclusion, demonstrating the progressive nature of probability calculations.