Problem 49 - Basic Statistics for Business and Economics with Student

Question

In a management trainee program at Claremont Enterprises, 80 percent of the trainees are female and 20 percent male. Ninety percent of the females attended college, and 78 percent of the males attended college. a. A management trainee is selected at random. What is the probability that the person selected is a female who did not attend college? b. Are gender and attending college independent? Why? c. Construct a tree diagram showing all the probabilities, conditional probabilities, and joint probabilities. d. Do the joint probabilities total 1.00 ? Why?

Step-by-Step Solution

Answer

a. 0.08 b. No, gender and attending college are not independent. c. See Step 5 explanation. d. Yes, they total 1.00 since they cover the whole probability space.

Step 1: Determine the Probability of a Female Trainee

The problem states that 80% of the trainees are female. So, the probability of selecting a female trainee is \( P(F) = 0.8 \).

Step 2: Determine the Probability of a Female Not Attending College

90% of the females attended college, leaving 10% who did not. Therefore, the probability of selecting a female who did not attend college is \( P(F' \mid F) = 0.10 \). Here, \( F' \) denotes females who did not attend college.

Step 3: Calculate Joint Probability for Female Not Attending College

The joint probability of selecting a female who did not attend college is calculated as the product of the probability of selecting a female and the probability that a female did not attend college: \( P(F \cap F') = P(F) \times P(F' \mid F) = 0.8 \times 0.10 = 0.08 \).

Step 4: Assess Independence Between Gender and College Attendance

Two events A and B are independent if \( P(A \cap B) = P(A) \times P(B) \). Let's calculate whether gender (Female) and attending college are independent events. \( P(C \mid F) = 0.9 \) and \( P(C \mid M) = 0.78 \), where C is attending college. Since \( P(C \mid F) eq P(C \mid M) \), gender and attending college are not independent.

Step 5: Construct the Tree Diagram (Explanation Only)

A tree diagram starts with the splitting of probabilities based on gender: 80% females, 20% males. Each gender splits further based on college attendance: 90% of females attended college (10% did not), and 78% of males attended college (22% did not). Label all proportion splits accordingly.

Step 6: Check Joint Probability Total

Calculate all joint probabilities: female attended college, female did not attend college, male attended college, and male did not attend college. Sum them: \( 0.72 + 0.08 + 0.156 + 0.044 = 1.00 \). The joint probabilities total 1.00 because this represents all possible outcomes of the entire sample space.

Key Concepts

Independent Events

In probability theory, independent events are events where the occurrence of one event does not affect the occurrence of another. This means the probability of one event happening does not change, regardless of whether the other event occurs or not. To check for independence mathematically, we use the formula:

• Two events, A and B, are independent if and only if \( P(A \cap B) = P(A) \times P(B) \).

In the provided exercise, the events are attending college and the gender of the trainee. To determine if these events are independent, compare the conditional probabilities.

• Probability of attending college given female: \( P(C \mid F) = 0.9 \)
• Probability of attending college given male: \( P(C \mid M) = 0.78 \)

Since these probabilities are not equal, gender and attending college are indeed not independent.

Joint Probability

Joint probability is the probability of both events occurring together. It's a key component in understanding how events are related, providing insight into outcomes that depend on two or more conditions. For the exercise, let's focus on finding the joint probability of certain events:

• For example, the joint probability of a trainee being female and not attending college is calculated as \( P(F \cap F') = P(F) \times P(F' \mid F) \).
• Given the values in the exercise, this is \( 0.8 \times 0.1 = 0.08 \).

Such calculations can help determine the probability of various combinations of events and are useful to analyze the overall behavior of different scenarios of trainees.

Conditional Probability

Conditional probability refers to the probability of an event occurring given that another event has already occurred. It is denoted as \( P(A \mid B) \), which means the probability of A given B. This concept is crucial when considering dependent events. In the problem, we explore the probabilities of college attendance conditioned on gender:

• Probability of a female attending college: \( P(C \mid F) = 0.9 \)
• Probability of a male attending college: \( P(C \mid M) = 0.78 \)

These conditional probabilities offer insights into how likely attendance is when considering gender, highlighting any potential dependencies between the events.

Tree Diagram

A tree diagram is a powerful visual tool used to display probabilities, especially useful in setups involving multiple stages or branches. They simplify the breakdown of complex problems, showcasing the sequence of events and associated probabilities. To construct a tree diagram for the exercise, follow these steps:

• Start by splitting based on gender: 80% female and 20% male.
• For each gender, further split into college attendance: for females, 90% attended college while 10% did not; for males, 78% attended while 22% did not.

Each branch represents a path through the scenario, and labeling probabilities along the branches helps illustrate the entire problem space. Summing all joint probabilities from the tree (female attended, female not attended, male attended, male not attended) will verify that they total to 1, reflecting the total probability of all possible outcomes.