Mathematics for Business and Economics: Conditional Probability

Mathematics for Business and Economics: Conditional Probability

Recap Section 4.5

• Elementary Rules:

  • $0 \le P(E) \le 1$

  • $P(S) = 1$ (Probability of the sample space is 1)

  • $P(\emptyset) = 0$ (Probability of the impossible event is 0)

• Union Rule: For any two events E and F,
  $P(E \cup F) = P(E) + P(F) - P(E \cap F)$

• Mutually Exclusive Events: If E and F are mutually exclusive (they cannot occur at the same time, meaning $E \cap F = \emptyset$),
  $P(E \cup F) = P(E) + P(F)$

• Complement Rule: For any event E,
  $P(E^c) = 1 - P(E)$

This Lecture's Content

This lecture focuses on:

• Defining conditional probability

• The Product Rule

• Using probability trees

• Understanding independent events

Conditional Probability: Motivation Example (1)

Scenario: If $1\%$ of UConn students are student-athletes, and $100\%$ of these student-athletes live in the United States.

Questions:

1. What is the probability of meeting a UConn student who lives in the United States AND is a student-athlete?

2. What is the probability of meeting a UConn student who lives in the United States GIVEN THAT that student is a student-athlete?

Solution:

• Let E: A student lives in the United States.

• Let F: A student is a student-athlete.

1. $P(E \cap F)$: This is the probability that a student is both a student-athlete AND lives in the United States. Since all student-athletes live in the USA, this is simply the probability of being a student-athlete.
   $P(E \cap F) = P(F) = 1\%$

2. $P(E \text{ given } F)$: This is the probability that a student lives in the United States, given they are a student-athlete. Since ALL student-athletes live in the USA, this probability is $100\%$.

   • Reasoning: If we know for certain a student is a student-athlete, and all student-athletes live in the USA, then it is certain that this student lives in the USA.

   • This demonstrates that more information leads to a better understanding of an event, and probabilities are