Problem 56 - Algebra and Trigonometry

Question

According to the Girl Scouts of America, \(19 \%\) of all Girl Scout cookies sold are Samoas/Caramel ookies deLites. If a box of Girl Scout cookies is selected at random, what is the probability that it does not contain Samoas/Caramel deLites?

Step-by-Step Solution

Answer

The probability is 0.81.

Step 1 - Understand the problem

Identify what you are given and what you need to find. You are given that 19% of all Girl Scout cookies sold are Samoas/Caramel deLites. You need to find the probability that a randomly selected box of Girl Scout cookies does not contain Samoas/Caramel deLites.

Step 2 - Express the given probability

Convert the given percentage into a probability. The problem states that 19% of the cookies are Samoas/Caramel deLites, which can be expressed as a probability of 0.19.

Step 3 - Calculate the complementary probability

To find the probability that a randomly selected box does not contain Samoas/Caramel deLites, subtract the probability that it does contain them from 1. The formula is: \[ P(\text{Not Samoas/Caramel deLites}) = 1 - P(\text{Samoas/Caramel deLites}) \]

Step 4 - Substitute and compute

Substitute the given probability into the formula and compute the result. \[ P(\text{Not Samoas/Caramel deLites}) = 1 - 0.19 = 0.81 \]

Key Concepts

Understanding Basic Probability

Probability helps us quantify the likelihood of an event occurring. Imagine the event as picking a girl scout cookie from a box. If 19% of all Girl Scout cookies are Samoas/Caramel deLites, we express this probability as 0.19 in decimal form. Basic probability ranges from 0 (impossible event) to 1 (certain event). For the given problem, the probability of picking a Samoas/Caramel deLites cookie is 0.19, highlighting that under random conditions, there's a 19% chance of selecting such a cookie. Remember, converting percentages to decimals is essential for calculations in probability.

Applying the Complement Rule

The complement rule in probability is used to find the probability of the opposite event occurring. When we talk about complements, we are essentially looking for events that 'complete' each other. For instance, if the probability of an event A happening is P(A), then the probability of A not happening, denoted as P(A'), is given by 1 - P(A). In our exercise, we are asked to find the probability that a random box of cookies **does not** contain Samoas/Caramel deLites. Using the complement rule:
P(Not Samoas/Caramel deLites) = 1 - P(Samoas/Caramel deLites).
This simplifies to 1 - 0.19 = 0.81. Hence, there is an 81% probability that a randomly selected box will not contain Samoas/Caramel deLites.

Converting Percentages to Probabilities

Percentage conversion is crucial for working with probabilities accurately. Percentages often represent probabilities as parts of 100. To use these in probability calculations, convert the percentages into decimals. This is done by dividing the percentage by 100. For example, 19% becomes 0.19 (19 ÷ 100). This conversion helps streamline calculations. In our example, knowing that 19% of cookies are Samoas/Caramel deLites directly translates into a probability of 0.19 for such cookies being picked, making probability calculations straightforward and efficient.