1
“Or” means add — but count nothing twice.
The king of hearts is in both lists, so it was counted twice.
4 + 13 − 1 = 16
Forgetting the overlap
4 + 13 = 17 counts the king of hearts as two different cards. There are only 16
cards that are a king or a heart.
With numbers, the overlap is the multiples of both, which are the multiples of the LCM. From 1 to 40, multiples of 4 or of 10: 10 + 4 − 2 = 12, since only 20 and 40 are multiples of both (multiples of 20, not of 4 × 10 = 40). So P = 1240 = 310.
2
Turn the count into a probability.
P =
1652
=
413
≈ 0.31
P(king or heart) = 413
The rule
P(A or B) = P(A) + P(B) − P(A and B)
3
When the overlap is empty.
Two events that cannot both happen are mutually exclusive.
king or heart
can overlap
subtract 1
king or queen
cannot overlap
subtract nothing
P(king or queen) =
452
+
452
=
213
No card is both a king and a queen, so the overlap is zero and the formula simplifies.
4
“And” means multiply.
Two kings in a row, drawn without replacement.
first card: 452
4 kings in the deck
second card: 351
one king and one card gone
452 ×
351
= 122652
multiply straight across
Both numbers change
Without replacement, the top and the bottom shrink. Using 452 twice
would be the answer only if the first card went back in.
5
The two words, side by side.
| Word |
Do |
Watch for |
| or | add | subtract the overlap |
|---|
| and | multiply | did the deck shrink? |
|---|
| at least one | 1 − P(none) | far quicker backwards |
|---|
“Or” answers get bigger. “And” answers get smaller.
That is a free sanity check: if your “and” answer came out larger than either
piece, something has gone wrong.
6
Drawing several at once: count with combinations.
all pairs: 9C2 = 36
equally likely
P(same colour) = 10 + 636 = 1636 = 49
red OR blue: the cases add
Only one colour
1036 is both red. “Same colour” means red-red or blue-blue, and the two cases cannot happen together, so they simply add.
7
Variations: probabilities instead of counts, “exactly one”, three draws.
Given as probabilities. P(A) = 0.6, P(B) = 0.5 and P(A and B) = 0.3. Find P(A or B), P(neither) and P(exactly one).
P(A or B) = 0.6 + 0.5 − 0.3 = 0.8
the same rule, no deck needed
P(neither) = 1 − 0.8 = 0.2
everything outside “or”
P(exactly one) = 0.8 − 0.3 = 0.5
“or” minus “both”
What is different: no outcomes to count. Picture the two-circle Venn diagram with 0.3 in the middle, 0.3 in A only, 0.2 in B only and 0.2 outside; they add to 1.
“Exactly one” is not “or”
0.8 includes the 0.3 who are in both. “Exactly one” removes them: P(A) + P(B) − 2P(A and B).
Three draws. Three cards are drawn without replacement. What is the probability that all three are face cards (jack, queen or king)?
1252 × 1151 × 1050
12 face cards; top and bottom both shrink each draw
= 1320132 600
multiply across
= 111105
divide top and bottom by 120
What is different: one more factor, the same habit. Cancel before multiplying: 1252 = 313, 1050 = 15, and 313 × 1151 × 15 = 333315 = 111105.
8
“Different” happens in two orders.
black then white: 45 × 15 = 425
one order
white then black: 15 × 45 = 425
the other order
P = 425 + 425 = 825
the orders cannot both happen: add
Check: same colour is (45)² + (15)² = 1725, and 1725 + 825 = 1 ✓.
Counting one order
425 is black-then-white only. Whenever the question does not fix which comes first, list both orders.
On the exam
Expect “or” with an overlap (hearts or face cards, multiples of 3 or of 5), “and” across two draws, or a selection made all at once that calls for combinations. Before adding, ask whether the two events can happen together; before multiplying, ask whether the first draw is put back. Ten-second check: P(event) + P(not event) = 1.
✓
Check: count the actual cards.
hearts
13 cards
including the king
other kings
3 cards
spades, clubs, diamonds
total
13 + 3 = 16
matches ✓
sensible?
1221 is tiny
two kings should be rare ✓
Remember the flow
Or · add · minus overlap | And · multiply
Ask whether the first item goes back.
