Questions: Probability Rules: Addition, Multiplication, and Complement

Question 1

A card is drawn from a standard 52-card deck. P(red) = 26/52, P(face card) = 12/52, P(red AND face card) = 6/52. What is P(red OR face card)?

• A. 38/52 — add P(red) and P(face card) directly
• B. 32/52 — subtract the overlap to avoid double-counting
• C. 20/52 — subtract P(red AND face card) from each term separately
• D. 6/52 — use the intersection since both conditions must hold

Answer: (B)

P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 26/52 + 12/52 − 6/52 = 32/52 = 8/13. The subtraction corrects for double-counting: the 6 red face cards were included in both P(red) and P(face card), so subtracting P(A ∩ B) once restores the correct count. Option A is the most common error — forgetting the subtraction and getting 38/52, as if red and face card were mutually exclusive.

Question 2

A bag contains 3 red and 7 blue marbles. You draw two marbles without replacement. A student calculates P(both red) = (3/10) × (3/10) = 9/100. What error did they make?

• A. They should have added the probabilities rather than multiplied them
• B. They applied the simplified multiplication rule P(A∩B) = P(A)·P(B) without verifying independence; draws without replacement are not independent
• C. They used the wrong sample space — there are only 2 marbles drawn, not 10
• D. The calculation is correct because each draw is a random event

Answer: (B)

P(A∩B) = P(A)·P(B) holds only when A and B are independent. Drawing without replacement makes the events dependent: after drawing one red marble, only 2 reds remain out of 9 total. The correct calculation is P(both red) = (3/10) × (2/9) = 6/90 = 1/15 ≈ 0.067, not 9/100 = 0.09. The simplified form tempts students because it's easier to apply, but it only works when independence has been confirmed.

Question 3

When two events are mutually exclusive, P(A ∪ B) = P(A) + P(B), with no subtraction needed, because mutually exclusive events share no outcomes.

• T. True
• F. False

Answer: True

Mutually exclusive events have P(A ∩ B) = 0 — they cannot both occur. In the addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B), subtracting 0 changes nothing. So the simpler form P(A ∪ B) = P(A) + P(B) is valid for mutually exclusive events. However, mutual exclusivity is a special case — for overlapping events, dropping the subtraction term is a significant error.

Question 4

If P(A) = 0.4 and P(B) = 0.5, then P(A ∩ B) = 0.2, because the multiplication rule gives P(A∩B) = P(A) × P(B).

• T. True
• F. False

Answer: False

P(A) × P(B) = P(A ∩ B) only when A and B are independent. Without knowing whether A and B are independent, you cannot calculate their joint probability from their individual probabilities alone. P(A ∩ B) could range anywhere from 0 (if mutually exclusive) to 0.4 (if A is a subset of B). Applying the simplified multiplication rule without confirming independence is one of the most common errors in probability.

Question 5

Why does the addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B) subtract the intersection, and when is it valid to drop that term?

Model answer: The subtraction corrects for double-counting. Any outcome in A ∩ B gets counted once in P(A) and again in P(B), so adding P(A) + P(B) counts the overlap twice. Subtracting P(A ∩ B) once removes the extra count, giving the correct probability of the union. It is valid to drop the subtraction term only when A and B are mutually exclusive — that is, when P(A ∩ B) = 0 — because there is no overlap to correct for.

The Venn diagram makes this concrete: the area of two overlapping circles equals the area of circle A plus the area of circle B minus the overlapping region (otherwise counted twice). For non-overlapping circles, there is no overlap to subtract. Recognizing whether events overlap is the first step in any union probability problem.