Welcome to Sections 4-3 to 4-6!

Hello Mathletes! Get ready to explore some exciting concepts in probability. These sections build upon your foundational knowledge and introduce powerful tools for analyzing events and their relationships. Let's break it down:

4-3: Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It's like saying, "What's the chance of this happening, knowing that that has already happened?"

Definition 4.6: Conditional Probability

The probability that event $B$ occurs given that event $A$ occurs is called a conditional probability. It is denoted $P(B | A)$, which is read "the probability of $B$ given $A$." We call $A$ the given event.

Formula 4.4: The Conditional Probability Rule

If $A$ and $B$ are any two events with $P(A) > 0$, then:

$$P(B | A) = \frac{P(A \& B)}{P(A)}$$

Example: Rolling a Die

Consider rolling a balanced die once. Let $F$ be the event a 5 is rolled, and $O$ be the event the die comes up odd.

  • $P(5 | odd) = \frac{1}{3}$ (The probability of rolling a 5, given that the outcome is odd)
  • $P(5) = \frac{1}{6}$ (The probability of rolling a 5)
  • $P(odd) = \frac{3}{6} = \frac{1}{2}$ (The probability of rolling an odd number)

Example: Marital Status and Gender

Let's look at the probability of an adult being divorced, given they are male. Using the provided table, we can determine this:

Let $A$ be the event that the adult is male, and $B$ be the event that the adult is divorced.

  • $P(B|A) = \frac{P(A \& B)}{P(A)} = \frac{0.044}{0.485} \approx 0.091$

4-4: Independence

Two events are independent if the occurrence of one does not affect the probability of the other. Think of it like flipping a coin twice – the outcome of the first flip doesn't change the odds of the second flip.

Definition 4.7: Independent Events

Event $B$ is said to be independent of event $A$ if $P(B | A) = P(B)$.

Formula 4.6: The Special Multiplication Rule (for Two Independent Events)

If $A$ and $B$ are independent events, then:

$$P(A \& B) = P(A) \cdot P(B)$$

4-5: Multiplication Rule

The multiplication rule helps us find the probability of two events both happening.

Formula 4.5: The General Multiplication Rule

If $A$ and $B$ are any two events, then:

$$P(A \& B) = P(A) \cdot P(B | A)$$

Example: U.S. Congress

What is the probability that a randomly selected member of the 113th Congress is a Democratic senator, given that 18.7% are senators and 53% of senators are Democrats?

$P(D \& S) = P(S) \cdot P(D | S) = 0.187 \times 0.530 = 0.099 \approx 9.9\%$

4-6: Counting Rules, Permutations and Combinations

This section introduces methods to count the number of possible outcomes in different scenarios. This includes the definition of factorials.

Definition 4.8: Factorials

The product of the first $k$ positive integers (counting numbers) is called $k$ factorial and is denoted $k!$. In symbols,

$$k! = k \cdot (k - 1) \cdot ... \cdot 2 \cdot 1$$

We also define $0! = 1$.

Formula 4.10: The Permutations Rule

The number of possible permutations of $r$ objects from a collection of $m$ objects is given by the formula:

$$_{m}P_{r} = \frac{m!}{(m - r)!}$$

Formula 4.12: The Combinations Rule

The number of possible combinations of $r$ objects from a collection of $m$ objects is given by the formula:

$$_{m}C_{r} = \frac{m!}{r!(m - r)!}$$

Keep practicing, and you'll master these concepts in no time! Good luck!