Rolling Two Dice (Independent Events)

What is the probability of rolling a 6 on the first die AND a 6 on the second die?

Identify the events: Event A: First die shows 6. Event B: Second die shows 6. = Two separate events

Check if independent: The result of the first die does not affect the second die. = Independent events

Find individual probabilities: $P(A) = \frac{1}{6}$ and $P(B) = \frac{1}{6}$ = Each die has 6 faces

Apply the multiplication rule: $P(A \text{ and } B) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$ = $\frac{1}{36}$

Answer: The probability of rolling double sixes is $\frac{1}{36}$ or about 2.78%.

Drawing Cards Without Replacement (Dependent Events)

A bag contains 5 red marbles and 3 blue marbles. You draw two marbles without replacement. What is the probability of drawing two red marbles?

Find P(first red): There are 5 red out of 8 total marbles. = $P(\text{1st red}) = \frac{5}{8}$

Find P(second red | first red): After drawing one red, there are 4 red out of 7 remaining. = $P(\text{2nd red}|\text{1st red}) = \frac{4}{7}$

Apply dependent probability formula: $P(\text{both red}) = \frac{5}{8} \times \frac{4}{7}$ = Multiply the probabilities

Calculate the result: $\frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$ = $\frac{5}{14}$

Answer: The probability of drawing two red marbles is $\frac{5}{14}$ or about 35.7%.

Either/Or Probability (Mutually Exclusive)

What is the probability of rolling a 3 OR a 5 on a single die?

Identify the events: Event A: Rolling a 3. Event B: Rolling a 5. = Two separate outcomes

Check if mutually exclusive: You cannot roll both 3 and 5 at the same time on one die. = Mutually exclusive

Find individual probabilities: $P(3) = \frac{1}{6}$ and $P(5) = \frac{1}{6}$ = Each outcome is equally likely

Apply the addition rule: $P(3 \text{ or } 5) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$ = $\frac{1}{3}$

Answer: The probability of rolling a 3 or a 5 is $\frac{1}{3}$ or about 33.3%.

Mistake: Adding probabilities when you should multiply

Why: When finding the probability of A AND B happening, you must multiply. Adding is only for A OR B (mutually exclusive events).

Correct: AND = multiply: $P(A \text{ and } B) = P(A) \times P(B)$. OR = add: $P(A \text{ or } B) = P(A) + P(B)$

Mistake: Treating dependent events as independent

Why: When one event affects another (like drawing without replacement), the probabilities change. Using the original probability gives wrong answers.

Correct: For dependent events, recalculate the probability after the first event. The total and favorable outcomes both change.

Mistake: Forgetting to simplify fractions

Why: The answer $\frac{20}{56}$ is correct but not fully simplified.

Correct: Always simplify: $\frac{20}{56} = \frac{5}{14}$. Find the GCF and divide both numerator and denominator.

Password Security

Banks and websites use compound probability to measure password strength.

A 4-digit PIN has 10 choices per digit. The probability of guessing it randomly is $\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10} = \frac{1}{10000}$.

Medical Testing

Doctors use compound probability to evaluate test accuracy.

If a medical test is 98% accurate, the probability of two independent tests both being wrong is $0.02 \times 0.02 = 0.0004$ or 0.04%.

Q: How do I know if events are independent or dependent?

A: Ask yourself: Does the first event change the situation for the second event? If drawing a card and NOT replacing it, the events are dependent because the deck changes. If flipping a coin twice, the events are independent because each flip has no effect on the next.

Q: Why do we multiply for AND but add for OR?

A: Think of AND as narrowing down possibilities - both conditions must be met, so the probability gets smaller (multiplication makes it smaller). OR expands possibilities - either condition works, so we combine the chances (addition makes it bigger).

Q: What does P(B|A) mean?

A: P(B|A) means 'the probability of B given that A has already happened.' It is the conditional probability - how likely is B after A occurs? For example, P(second red | first red) is the probability of drawing a second red marble after already drawing a red marble.