Numerical aptitude · Counting and chance
Probability, made easy
💡 The idea
Probability = good results ÷ all results (when every result is equally likely). Draw all the results, colour the good ones, count.
🎲 one dice: 6 results→even: 2, 4, 6 → 3→3/6 = 1/2
🧭 The method: 3 steps
- List (or picture) all the equally likely results.
- Count the ones you want.
- Divide and simplify. “At least one”? Work out 1 − (none).
🎬 Every kind of question, in pictures
Pick a kind. Guess the answer first, then watch the picture work it out. Tap any step to see it in the picture.
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✏️ Worked examples
Example 1. Two dice. Total 7?
- 6 × 6 = 36 results
- (1,6) (2,5) (3,4) (4,3) (5,2) (6,1) → 6
- 6/36 = 1/6
Example 2. 3 red, 2 blue. Pick 2 without putting back. Both red?
- First red: 3/5
- Second red: 2/4
- 3/5 × 2/4 = 3/10
⚠️ Where people go wrong
- Totals of two dice are NOT equally likely (7 is the most common).
- Without putting back, the second pick has one fewer.
- “King or heart”: do not count the king of hearts twice.
📌 Remember
- P = good ÷ all
- P(not) = 1 − P
- “And” → multiply
- Two dice: 36 results