3.2: Independent and Mutually Exclusive Events

Foundations of Probability and Sample Space

• The sample space is defined as the collection of outcomes that can occur from an experiment.

• When an experiment consists of equally likely outcomes, the probability is calculated by counting the outcomes that satisfy the event.

Conditional Probability

• Conditional probability is used to describe the probability of event $A$ occurring under the condition that event $B$ is met.

• The formula for conditional probability is:

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

• The conditional probability $P(A|B)$ is NOT the same as $P(B|A)$.

• Conditional Probability Example:

  • Scenario: Flip $3$ coins.

  • Problem: What is the probability that at least two heads are flipped if the last flip has to be Tails? Use the conditional probability formula to calculate.

  • Equivalence: This is the same as asking for $P(HHT)$. Calculating it both ways produces equal answers.

Independent Events

• Two events are independent if the knowledge of one event does not affect the likelihood of the other event.

• Mathematical conditions for independence:

$P(A|B) = P(A)$

$P(B|A) = P(B)$

• Product Rule for Independent Events:

  • Being independent also means that:

$P(A \cap B) = P(A)P(B)$

• Independence in Chance Experiments:

  • Most chance experiments (such as flipping coins or rolling dice) are independent from each other if the experiment remains unchanged from one iteration to the next.

• Drawing Items With vs. Without Replacement:

  • An experiment with drawing items (like cards) can be independent if the items are drawn with replacement, meaning they are put back and mixed (shuffled) before drawing again.

  • Drawing without replacement can occur as well, but that experiment is no longer independent since the mixture has changed from one draw to the next.

• Card Drawing Examples:

  • Example 1 (Without Replacement): Draw two cards without replacement. What is the probability that the second card is "Ace of Hearts" if the first card is "Ace of Hearts?"

  • Example 2 (With Replacement): Draw two cards with replacement. What is the probability that the second card is "Ace of Hearts" if the first card is "Ace of Hearts?"

Mutually Exclusive Events

• Two events are mutually exclusive if it is impossible for both events to occur simultaneously.

• Mathematical condition for mutually exclusive events:

$P(A \cap B) = 0$

• Mutually Exclusive Example:

  • Scenario: Roll a fair six-sided die.

  • Problem: What is $P(\text{odd} \cap \text{at least } 6)$?

• Breaking Down Complex Probabilities:

  • Mutually exclusive events can be helpful when breaking down the probabilities of more complicated events.

  • Example: Getting an "odd" result on a die can be thought of as three events: "getting $1$ or getting $3$ or getting $5$", which are all mutually exclusive.

• Addition Rule for Mutually Exclusive Events:

  • When events are mutually exclusive, the probability that any one of them occurs is equal to the sum of their individual probabilities:

$P(A \cup B) = P(A) + P(B)$

• Addition Rule Example:

  • Scenario: Roll a fair six-sided die.

  • Problem: Compute $P(\text{odd})$ and then compute $P(1 \cup 3 \cup 5)$.