Sample spaces and the addition rule.
P(A or B) = P(A) + P(B) − P(A and B). The standard deck of 52 cards is the canonical playground.
Draw one card from a shuffled deck. What's the probability the card is red OR a face card? You know P(red) = 26/52 = 1/2 from L1 (half the deck is red). You know P(face) = 12/52 because there are 12 face cards (J, Q, K in each of four suits). So is P(red OR face) = 26/52 + 12/52 = 38/52?
No. That answer double-counts the six cards that are both red AND a face: J♥, Q♥, K♥, J♦, Q♦, K♦. Each one gets counted once when you tally "red" and a second time when you tally "face." To get the right count, you have to subtract the overlap once.
That correction — "don't count the overlap twice" — is the addition rule. It's the formula that handles every OR question in probability, and it's where the standard 52-card deck earns its place as the training ground for the topic.
Add the parts; subtract the overlap.
P(A or B) = P(A) + P(B) − P(A and B)
The first two terms count every outcome in A and every outcome in B. But outcomes in BOTH — the overlap — get counted by each of the first two terms, so they appear twice in the sum. Subtracting P(A and B) once removes the duplicate, leaving each outcome counted exactly once.
When A and B share no outcomes (the overlap is empty), P(A and B) = 0 and the formula reduces to the simpler P(A) + P(B). Events with no overlap are called mutually exclusive — rolling a 3 and rolling a 5 on the same die, for instance: they can't both happen at once.
The deck split two ways. Red contains 26 cards; Face contains 12 cards; the intersection (red AND face) contains 6 cards — the J/Q/K of hearts and diamonds. The total "red OR face" region is the two circles' union: 26 + 12 − 6 = 32 cards.
The addition rule for two events A and B states P(A or B) = P(A) + P(B) − P(A and B). When A and B are mutually exclusive (they cannot both happen), the overlap term is zero and the rule simplifies to P(A or B) = P(A) + P(B).
Words you'll see on ALEKS
- Union (A or B) The event "A happens OR B happens (or both)." Includes every outcome that's in either set. In set notation: A ∪ B.
- Intersection (A and B) The event "A happens AND B happens at the same time." Includes only the overlap. In set notation: A ∩ B. The intersection is what the addition rule subtracts.
- Mutually exclusive Two events are mutually exclusive when they cannot both occur on the same trial. Rolling a 3 and rolling a 5 on the same die roll. Their intersection is empty, so P(A and B) = 0 and the addition rule simplifies.
- Standard 52-card deck 52 cards in 4 suits (♠ ♣ ♥ ♦), 13 cards per suit (A, 2-10, J, Q, K). Hearts and diamonds are red (26 cards); spades and clubs are black (26 cards). Face cards are J, Q, K (12 total, three per suit). The canonical deck shows up in every ALEKS probability review.
- Double-counting The mistake of including the same outcome twice. Happens when you compute P(A or B) as P(A) + P(B) without subtracting the overlap. Always subtract P(A and B) — the inclusion-exclusion rule is the fix.
P(face), P(red), P(face or red).
"Suppose one card is drawn at random from a standard deck of 52 cards. Find: (a) P(face), (b) P(red), (c) P(face OR red). Write your answers as fractions."
(a) P(face) — count the face cards.
Face cards are J, Q, K in each of four suits: 3 × 4 = 12 face cards out of 52 total.
(b) P(red) — count the red cards.
Two of the four suits are red (hearts and diamonds): 2 × 13 = 26 red cards.
(c) P(face OR red) — addition rule.
Apply the formula, but first identify the overlap. The cards that are both face AND red are the J/Q/K of hearts and diamonds: 6 cards. So P(face AND red) = 6/52.
= 12/52 + 26/52 − 6/52
= 32/52 = 8/13
Sanity check: the union has 32 cards total — 26 red cards plus 6 black face cards (J♠, Q♠, K♠, J♣, Q♣, K♣) that are face but not red. 32/52 = 8/13.
→ subtract the overlap onceThree OR questions.
A fair six-sided die is rolled. What is P(roll a 2 OR roll a 5)?
One card is drawn from a standard 52-card deck. What is P(king OR heart)?
Of 30 students, 18 play soccer, 12 play basketball, and 7 play both. One student is picked at random. What is P(soccer OR basketball)?
Three fast questions before you move on.
Q1. Two events A and B are mutually exclusive. Which formula computes P(A or B)?
Q2. A card is drawn from a standard deck. What is P(spade or club)?
Q3. If P(A) = 0.4, P(B) = 0.5, and P(A AND B) = 0.2, what is P(A OR B)?
OR questions, anywhere they appear.
The addition rule shows up every time you want to combine two events with OR. Polls ("voters who lean left OR are under 30"), epidemiology ("patients with diabetes OR high blood pressure"), insurance ("claim filed for collision OR theft") — the structure is identical, just larger sample spaces.
The single most important habit: before you add, ask whether the overlap is zero. If the events are mutually exclusive, you can skip the subtraction. If they're not, subtract the intersection once to avoid double-counting.
Next: Lesson 4 introduces permutations and combinations — what to do when the counting principle's "independent steps" assumption breaks down because each choice removes an option from the next step.
Continue to Lesson 04Different angle? Need another rep? These are optional — tap any that look helpful.
Addition rule for probability
Sal derives P(A or B) = P(A) + P(B) − P(A and B) using Venn diagrams and explains why we subtract the overlap. Crisp foundational treatment of the formula.
Probability with playing cards and Venn diagrams
The exact 'P(red card or king)' worked example, with both the addition rule and a Venn diagram showing the overlap subtraction step explicitly.