Question

Despite reports that dark chocolate is beneficial to the heart, \(47 \%\) of adults still prefer milk chocolate to dark chocolate. \({ }^{11}\) Suppose a random sample of \(n=5\) adults is selected and asked whether they prefer milk chocolate to dark chocolate. a. What is the probability that all five adults say that they prefer milk chocolate to dark chocolate? b. What is the probability that exactly three of the five adults say they prefer milk chocolate to dark chocolate? c. What is the probability that at least one adult prefers milk chocolate to dark chocolate?

Short answer

Answer: The probability that all five adults prefer milk chocolate to dark chocolate is approximately \(0.0401\), that exactly three of the five adults prefer milk chocolate to dark chocolate is approximately \(0.3623\), and that at least one adult prefers milk chocolate to dark chocolate is approximately \(0.9579\).

Step-by-step solution

a. Probability that all five adults prefer milk chocolate to dark chocolate

We need to find the probability that all 5 adults in the sample prefer milk chocolate over dark chocolate (\(k = 5\)). We will use the binomial probability formula: \(P(5) = \begin{pmatrix} 5\\ 5\end{pmatrix} \cdot (0.47)^5 (1-0.47)^{5-5}\) Calculate the probability: \(P(5) = 1 \cdot (0.47)^5 (0.53)^0 = (0.47)^5\) \(P(5) \approx 0.0401\) The probability that all five adults prefer milk chocolate to dark chocolate is approximately \(0.0401\).

b. Probability that exactly three of the five adults prefer milk chocolate to dark chocolate

We need to find the probability that exactly 3 adults in the sample prefer milk chocolate over dark chocolate (\(k = 3\)). We will use the binomial probability formula: \(P(3) = \begin{pmatrix} 5\\ 3\end{pmatrix} \cdot (0.47)^3 (1-0.47)^{5-3}\) Calculate the probability: \(P(3) = 10 \cdot (0.47)^3 (0.53)^2 \approx 0.3623\) The probability that exactly three of the five adults prefer milk chocolate to dark chocolate is approximately \(0.3623\).

c. Probability that at least one adult prefers milk chocolate to dark chocolate

We need to find the probability that at least one adult prefers milk chocolate (\(k \geq 1\)). To do this, we can find the complementary probability that none of the adults prefer milk chocolate (\(k = 0\)) and subtract it from 1. We will use the binomial probability formula: \(P(0) = \begin{pmatrix} 5\\ 0\end{pmatrix} \cdot (0.47)^0 (1-0.47)^{5-0}\) Calculate the probability: \(P(0) = 1 \cdot 1 \cdot (0.53)^5 \approx 0.0421\) Now, find the complementary probability: \(P(K \geq 1) = 1 - P(0) = 1 - 0.0421\approx 0.9579\) The probability that at least one adult prefers milk chocolate to dark chocolate is approximately \(0.9579\).

Key concepts

Probability

Probability is a tool that helps us quantify uncertainty. It gives us a numerical value between 0 and 1, representing the likelihood of an event happening. For example, if an event has a probability of 1, it is certain to happen, while a probability of 0 indicates that the event is impossible. In the case of the chocolate preference example with adults, we use probabilities to determine the chances of different scenarios occurring during our survey.

In the exercise, we explore the probability of adults preferring milk chocolate over dark chocolate. We express these scenarios using a value derived from the ratio of individuals who prefer milk chocolate over the total number of surveyed adults. Calculating such a possibility helps in making informed decisions and predictions based on varying outcomes.

Random Sample

A random sample is a key concept in statistics. It ensures that every individual in a population has an equal chance of being selected for the study. In the context of the chocolate preference survey, obtaining a random sample of five adults helps depict an accurate representation of the whole population.

Random sampling aims to minimize biases and allow for manageable and reliable predictions. The random nature of sampling means that inherent personal preferences don't skew results, maintaining the study’s integrity. It helps avoid patterns that could misrepresent the broader population's chocolate preferences.

Complementary Probability

Complementary probability is a handy concept in probability, used to calculate the chance of an event not occurring. Simply put, if you have the probability of an event happening, the complementary probability helps you deduce the probability of the event not happening by subtracting it from 1.

In our exercise, we calculated the probability that zero adults prefer milk chocolate. To find the probability of at least one adult preferring milk chocolate, we used complementary probability. We subtracted the probability of none preferring it (i.e., all preferring dark chocolate) from 1, giving us the probability where at least one adult would opt for milk chocolate. This approach streamlines calculations involving the occurrence of at least one particular outcome, offering an efficient method to find such probabilities.

Binomial Probability Formula

The binomial probability formula serves as a vital tool in probability calculations, especially when dealing with binary outcomes like yes/no, success/failure, or in this case, preference for one type of chocolate over another. This formula is defined as: \[P(k) = \begin{pmatrix} n\k\end{pmatrix} \cdot p^k \cdot (1-p)^{n-k} \] where

• \(n\) is the sample size,
• \(k\) is the number of successful outcomes we are interested in,
• \(p\) is the probability of a single success (adult preferring milk chocolate),
• and \((1-p)\) is the probability of failure.

We used this formula when calculating the probabilities for different numbers of adults preferring milk chocolate in our sample. It's a versatile tool, perfect for scenarios where there are two potential outcomes, allowing researchers to calculate specific probabilities with precision.