Probability

Skills and methods in this topic, easy to hard

1. 1Certain, impossible and random eventsAll the outcomes: certain. None: impossible. Some: random

   MethodA die: "a 7" impossible, "less than 7" certain, "even" random

   Another way: The unluckiest case3 red, 4 blue: draw at least 4 + 1 = 5 to be sure of a red

2. 2What a probability means0 ≤ P ≤ 1; P(not) = 1 − P; experimental = times ÷ trials

   MethodP(rain) = 310, P(no rain) = 1 − 310 = 710

   Another way: From the counts120 rolls × 16 = 20 times

3. 3Equally likely outcomesP = favourable ÷ all outcomes, in simplest form

   Method3 red, 5 blue: P(red) = 3 ÷ 8 = 38

   Another way: One take away1 − 520 = 34

4. 4Relative frequency and expected frequencyRelative frequency = frequency ÷ trials; expected frequency = probability × trials

   Method46 ÷ 200 = 0.23; 0.4 × 300 = 120

   Another way: Against the theory180 ÷ 600 = 0.3, far more than 16

5. 5Listing outcomes in a tableTwo dice: 36 cells; count the favourable ones. The sums aren't equally likely

   MethodSum 8: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2), P = 536

   Another way: The diagonalsSum 8: 13 − 8 = 5 cells

6. 6Tree diagramsMultiply along a path; without replacement, one fewer the second time

   Method3R 2B, no replacement: 35 × 24 = 310

   Another way: Counting ordered outcomes(3 × 2) ÷ (5 × 4) = 620 = 310

7. 7Probability word problemsMultiples of 3 or 5: 10 + 6 − 2; take off the ones counted twice

   Method1 to 30: (10 + 6 − 2) ÷ 30 = 715

   Another way: Listing them3, 5, 6, 9, 10, 12, 15, 18, 20, 21, 24, 25, 27, 30: 14

8. 8Expected valueE = Σ value × probability: the average over many tries

   MethodSpinner $0, $0, $10, $30: E = 40 × 14 = $10

   Another way: Each outcome once, then the mean(0 + 0 + 10 + 30) ÷ 4 = 10

9. 9Fair games and expected gainExpected gain = expected prize − cost; 0 is fair

   Method24 × 16 − 5 = 4 − 5 = −1

   Another way: Picture 6 plays(24 − 6 × 5) ÷ 6 = −1

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

1. Example 1Certain, impossible and random events →

   An ordinary die is rolled once. Which of these is an impossible event?

   • A. Rolling a 7
   • B. Rolling an even number
   • C. Rolling a 6
   • D. Rolling a number less than 7

   Choose the right answer

   ▶Show the answer and working

   Answer: A. Rolling a 7

   A die shows only 1 to 6, never 7, so rolling a 7 can't happen.

2. Example 2Listing outcomes in a table →

   Two dice are rolled. The table lists the product of the two. The probability of a product greater than 5 is .

   Answer in simplest form

   ▶Show the answer and working

   Answer: 1318

   Of the 36 cells, 26 show more than 5: 2636 = 1318.

3. Example 3Fair games and expected gain →

   One play costs 5 dollars. A die is rolled: a 6 wins 24 dollars, anything else nothing. What is the expected gain from one play, in dollars (a loss as a negative number)?

   ▶Show the answer and working

   Answer: −1 dollars

   Expected prize = 24 × 16 = 4; expected gain = 4 − 5 = −1.