Answered

  1. $P(\text{not a snowmobile renter}) = \square\ \frac{952}{1526}$

There is a $\square\ 0.624$ probability that an environmental club member is not a snowmobile renter.


  1. What is $P(\text{environmental club member and not a snowmobile renter})$? $\square\ 0.193$ (Round to 3 decimal places)


  1. Use the general addition rule to find the probability that the person is not an environmental club member or is a snowmobile renter.



Let $A =$ person who is not an environmental club member

Let $B =$ person who is a snowmobile renter

Verified Answer :

  • We define the probability of being not an environmental member or a snowmobile renter as: \(P(E^c \cup S) = P(E^c) + P(S) - P(E^c \cap S)\).
  • Using the provided values, we derive: \(P(E) = \frac{0.193}{0.624} \approx 0.309\), \(P(E^c) \approx 0.691\), and \(P(S) \approx 0.376\).
  • We find the joint probability \(P(E^c \cap S) = P(S) - (P(E) - P(E \cap S^c)) \approx 0.260\).
  • Summing these values gives the final result: \(\boxed{0.807}\).

Explanation

  1. Understand the problem and gather data

To find the probability that a person is either not an environmental club member (\(E^c\)) or is a snowmobile renter (\(S\)), we use the general addition rule for probabilities:
\[P(E^c \cup S) = P(E^c) + P(S) - P(E^c \cap S)\]
First, we define our events and identify the given information:

  • Let \(E\) be the event that a person is an environmental club member.
  • Let \(S\) be the event that a person is a snowmobile renter.
  • We are given:
  • \(P(S^c) = \frac{952}{1526} \approx 0.62385\)
  • \(P(S^c | E) = 0.624\)
  • \(P(E \cap S^c) \approx 0.193\)
  1. Calculate necessary individual probabilities

Next, we calculate the individual probabilities needed for the addition rule:

  • Probability of being an environmental club member (\(P(E)\)):

Using the definition of conditional probability, \(P(S^c | E) = \frac{P(E \cap S^c)}{P(E)}\), we rearrange it to:
\[P(E) = \frac{P(E \cap S^c)}{P(S^c | E)} = \frac{0.193}{0.624} \approx 0.30929\]

  • Probability of NOT being an environmental club member (\(P(E^c)\)):

\[P(E^c) = 1 - P(E) = 1 - 0.30929 \approx 0.69071\]

  • Probability of being a snowmobile renter (\(P(S)\)):

\[P(S) = 1 - P(S^c) = 1 - \frac{952}{1526} \approx 1 - 0.62385 = 0.37615\]

  1. Determine the intersection probability \(P(E^c \cap S)\)

Now, we find the intersection probability \(P(E^c \cap S)\):

  • First, find the probability that a person is both an environmental club member and a snowmobile renter (\(P(E \cap S)\)):

\[P(E \cap S) = P(E) - P(E \cap S^c) = 0.30929 - 0.193 = 0.11629\]

  • Then, use the law of total probability for \(P(S)\) to find \(P(E^c \cap S)\):

\[P(S) = P(E \cap S) + P(E^c \cap S)\]
\[P(E^c \cap S) = P(S) - P(E \cap S) = 0.37615 - 0.11629 = 0.25986\]

  1. Calculate the final probability and conclude

Finally, substitute the calculated values into the general addition rule formula:
\[P(E^c \cup S) = P(E^c) + P(S) - P(E^c \cap S)\]
\[P(E^c \cup S) = 0.69071 + 0.37615 - 0.25986 = 0.807\]
The probability that a person is not an environmental club member or is a snowmobile renter is approximately \(0.807\).

Examples

This method is frequently used in market research and survey analysis. For instance, if a company wants to understand their customer base, they might calculate the overlap between different demographic groups or interests—such as knowing how many customers belong to a specific club versus how many rent their equipment—to optimize marketing strategies and product offerings. Using the addition rule ensures that individuals belonging to both groups aren't counted twice, providing an accurate measure of total reach.

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