Lesson 4-4 – Conditional Probability & Independent Events

These Q&A flashcards review definitions, rules, and illustrative examples for independent events, mutually exclusive events, conditional probability, and compound probability calculations from Lesson 4-4.

What is the definition of independent events in probability?

Two events are independent when the occurrence of one does not change the probability of the other occurring.

How does replacing an item (“with replacement”) affect independence?

With replacement keeps probabilities the same for each trial, so the events are independent.

How does NOT replacing an item (“without replacement”) affect independence?

Without replacement alters the remaining probabilities, so the events become dependent.

Are the events “flip a head on a coin” and “roll a 5 on a die” independent? Why?

Yes; the chance of rolling a 5 (1⁄6) is the same whether or not a head is flipped.

Why are “robbing a bank” and “going to prison” NOT independent?

Because robbing a bank greatly increases the probability of going to prison compared with not robbing a bank.

Can events be both independent and mutually exclusive?

No. If two events are mutually exclusive they cannot occur together, which forces dependence.

State the AND (intersection) rule for independent events.

P(A ∩ B) = P(A) · P(B).

State the AND (intersection) rule for dependent events.

P(A ∩ B) = P(A) · P(B | A).

State the OR (union) rule for any two events A and B.

P(A ∪ B) = P(A) + P(B) – P(A ∩ B).

When is the OR rule P(A ∪ B) = P(A)+P(B) used without subtraction?

When A and B are mutually exclusive; then P(A ∩ B)=0.

Write the formula for conditional probability.

P(B | A) = P(A ∩ B) / P(A), provided P(A) ≠ 0.

How can you prove two events A and B are independent using conditional probability?

Show that P(B | A) = P(B) (or equivalently P(A | B) = P(A)).

Define mutually exclusive (disjoint) events.

Events that cannot occur together in a single trial; their intersection is empty.

Give an example of mutually exclusive events with a die.

Rolling a 3 and rolling a 4 on the same single die roll.

Why do we subtract P(A ∩ B) in the OR rule?

Because the outcomes in the overlap are counted twice—once in P(A) and once in P(B)—so one copy must be removed.

In a 2×2 contingency table, how do you find P(grade | female)?

Divide the number of girls who choose grades by the total number of girls.

If P(girl)=0.484 and P(girl ∩ grades)=0.400, are gender and choice independent?

No, because P(grades | girl)=0.827 ≠ P(grades)=0.548 (for the whole sample).

Sock drawer: What rule is usually applied when the wording says “at least one”?

Use the complement rule: P(at least one) = 1 – P(none).

What is the probability of drawing two blue socks (4 blue, 5 gray, 3 red, no replacement)?

(4/12) · (3/11) = 12/132 ≈ 0.091 (9.09%).

What does P(stop at least once during 5 days) equal if P(red)=0.61 each day?

1 – (0.39)^5 ≈ 0.991 (99.1%).

Superbowl fans: If 65% root Rams, 46% Bengals, 29% both, what % root for at least one?

P(R ∪ B) = 0.65 + 0.46 – 0.29 = 0.82 or 82%.

Using the same data, what % root for neither team?

1 – 0.82 = 0.18 or 18%.

DUI tests: 78% breath, 36% blood, 22% both. What % receive some test?

0.78 + 0.36 – 0.22 = 0.92 or 92%.

DUI tests: What % receive only a blood test?

0.36 – 0.22 = 0.14 or 14%.

If two outcomes share the same sample space but have at least one common element, are they mutually exclusive?

No, because at least one outcome makes them overlap.

Die roll: Why are “multiple of 3” and “even number” not mutually exclusive?

Because 6 is both even and a multiple of 3, so the events overlap.

Traffic lights: Why can Monday’s and Tuesday’s light colors be treated as independent?

Each day’s light color is assumed to have the same probabilities and is not affected by previous days.

Give the joint probability of two independent red lights on consecutive days if P(red)=0.61.

0.61 × 0.61 = 0.3721 (37.21%).

Explain why independence involves two separate actions while mutual exclusivity involves one action.

Independence addresses whether one event in a sequence affects another; mutual exclusivity addresses whether two events can happen simultaneously in a single trial.

When testing independence, what conclusion do you draw if P(A ∩ B) = P(A)·P(B)?

The events A and B are independent because the joint probability equals the product of individual probabilities.