Organizing combined events
Tree diagrams: Show all paths.
Label branches with probabilities.
Final outcomes: multiply along path.
Probability tables: Organize outcomes in rows and columns.
Easier for many events.
Worked example
Bag: 3 red, 2 blue.
Draw 2 (no replace).
Find all P outcomes.
Solution
- First red (3/5): second red (2/4)=6/20
- First red (3/5): second blue (2/4)=6/20
- First blue (2/5): second red (3/4)=6/20
- First blue (2/5): second blue (1/4)=2/20
Final answer
RR=6/20, RB=6/20, BR=6/20, BB=2/20. Total=20/20=1.
IB-style question — probability from a two-way table
Of 100 households: 40 own a dog, and 25 of those also own a cat. Of the 60 without a dog, 10 own a cat.
(a) Find P(owns a cat).
Step by step
- Fill the table: dog&cat = 25, dog-only = 15, cat-only = 10, neither = 50. Total cats = 25 + 10 = 35.
Final answer
0.35.
Part (b)
(b) Find P(owns a cat | owns a dog).
Step by step
- Conditional — restrict to the 40 dog owners.
Final answer
0.625.
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"At least one": use the complement
Three independent free throws
Ana scores each free throw with probability 0.7, independently of the others. She takes three free throws.
(a) Find the probability that she scores all three.
Solution
- Independent: multiply.
Final answer
0.343.
Part (b)
(b) Find the probability that she misses at least one.
Solution
- "Misses at least one" is everything except "scores all three".
Final answer
0.657.
Part (c)
(c) Find the probability that she scores at least one.
Solution
- "None" means three misses, each with probability 1 − 0.7 = 0.3.
Final answer
0.973.
Why not add the cases?
"At least one score" in three throws is 7 different patterns. "None" is one pattern: one subtraction replaces seven products.
"At most one" — same trick?
Yes, when the opposite is short.
Example: two throws, at most one score = 1 − P(both score) = 1 − 0.49 = 0.51.
Exam tip: Write "1 − P(none)" before the numbers: it is the method mark.
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"Exactly … in total": list the cases
List every way, then add: A total over two independent games can be made in several ways. Write each way as a product, then add the products.
| Points in one game | 0 | 1 | 2 |
|---|---|---|---|
| Probability | 0.3 | 0.5 | 0.2 |
A total of 2 points in two games
In one round of a game, Leo wins 0, 1 or 2 points with the probabilities in the table. He plays two rounds, independently. Find the probability that he wins exactly 2 points in total.
Solution
- The ways to make 2: (0 then 2), (1 then 1), (2 then 0).
- Multiply along each way.
- Add the ways.
Final answer
0.37
Why is (0, 2) different from (2, 0)?
They are different results in different rounds. Each one is counted.
Check: (1, 1) happens in only one way, so it is counted once.
How many marks for the cases?
Usually one method mark per correct product, and one for adding them. Show each product.
Exam tip: Missing the reversed order — (2, 0) as well as (0, 2) — is the most common lost mark.
Two different experiments: list the sample space
A grid shows every equally likely pair: One experiment down the side, the other across the top. Every cell is one equally likely outcome.
| Coin \ Die | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Tails | 1 | 2 | 3 | 4 | 5 | 6 |
| Heads | 2 | 3 | 4 | 5 | 6 | 7 |
Coin then die
In a game, a fair coin is tossed and then a fair six-sided die is rolled. Tails: the score is the number on the die. Heads: the score is the number on the die plus 1.
(a) Find the probability that the score is 7.
Solution
- There are 2 × 6 = 12 equally likely pairs (the grid).
- A score of 7 needs heads AND a 6: one cell. Or multiply.
Final answer
1/12 ≈ 0.0833.
Part (b)
(b) Find the probability that the score is 4.
Solution
- A score of 4: tails and 4, or heads and 3 — 2 cells.
Final answer
1/6 ≈ 0.167.
Exam tip: Say which pairs give the score: "only heads and a 6 gives 7" is the method mark before the product.
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Without replacement: several picks from a group
Both numbers drop by one: Choosing people (or items) one after another means nobody is chosen twice. After each pick, the group of that kind AND the whole group are one smaller.
Choosing a team
A club has 30 members. 12 of them play chess. Members are chosen at random, one after another.
(a) Three members are chosen. Find the probability that all three play chess.
Solution
- Multiply three fractions; top and bottom each drop by 1.
Final answer
0.0542.
Part (b)
(b) Two members are chosen. Find the probability that both play chess or both do not play chess.
Solution
- Two ways to match: both chess, or both not (18 members do not play).
- Add the two products.
Final answer
0.503.
From a two-way table?
Take the group size and the grand total from the table, then use the same products.
Example: 25 of 80 applicants chose Art → both chose Art = (25/80) × (24/79).
With replacement instead?
The numbers stay the same each time.
Example: (12/30) × (12/30).
Exam tip: Write the full product of fractions before the decimal: one mark is for the product, one for the correct fractions.