Year 8 · Unit 2 Probability · Lesson 4 of 6

Count the ones you want, out of all of them

When every outcome is equally likely, the probability of an event is the outcomes you want divided by all the outcomes. The probability of it not happening is what is left up to 1.

Statistics and ProbabilityNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. How many outcomes are there when a coin is tossed three times?
    Answer8. 2 × 2 × 2 paths on the tree.
  2. Write 6/36 in its simplest form.
    Answer1/6. Divide the top and the bottom by 6.
  3. Write 0.2 as a percentage, and find 25% of 80.
    Answer20%, and 20. A quarter of 80 is 20.
Words used in this lesson
theoretical probability
the probability worked out from equally likely outcomes. For example: a die shows 5 with probability 1/6.
favourable outcome
an outcome that makes the event happen. For example: 8, 9 and 10 for “greater than 7”.
equally likely
each outcome has the same chance as every other. For example: the faces of a fair die.
complement
the event not happening. For example: not red.
fair
with no outcome favoured over any other. For example: a fair coin.

Equally likely outcomes, counted

Two fair dice are rolled and the numbers added. The totals run from 2 to 12. Before you open the figure, which total do you think comes up most often, and which least often?

With equally likely outcomes, every cell of the grid is as likely as every other. So a probability is a count: the cells that make the event happen (favourable outcomes) out of all the cells.

P(A) = favourable outcomes ÷ all outcomes, when every outcome is equally likely.

P(not A) = 1 − P(A). Event and not-event use up every outcome between them, so their probabilities add to 1.

Slide the total and read the count of shaded cells before the fraction. Then set the total you chose in your prediction and check.

3
2cells with this total, out of 36
2/36 = 1/18P(this total)
34/36 = 17/18P(not this total)

A total of 3 is on 2 of the 36 cells, (1, 2) and (2, 1): P = 2/36 = 1/18. The other 34 cells are P(not 3) = 34/36 = 17/18.

Show me the steps
  1. Total 4: 3 cells, (1, 3), (2, 2) and (3, 1). P = 3/36 = 1/12.
  2. Total 12: 1 cell, (6, 6). P = 1/36, and P(not 12) = 35/36.
  3. Total 7: 6 cells, one in every row. P = 6/36 = 1/6, and P(not 7) = 30/36 = 5/6.
  4. For any total, the shaded cells and the unshaded cells together make all 36, so P and P(not) always add to 1.

Count equally likely outcomes, not results. There are 11 totals, but they are not equally likely, so a total of 7 is not 1/11. It sits on 6 of the 36 equally likely pairs, so it is 6/36 = 1/6. Before dividing, check that every item you are counting is as likely as any other. Rafa's “one of each” from Lesson 3 is 2 of 4 equally likely outcomes, so P = 1/2, not 1/3.

Worked examples

A bag holds 3 red and 5 yellow counters. Find P(red) and P(not red).

  1. 3 + 5 = 8 counters in all, each equally likely to be picked.
  2. P(red) = 3/8.
  3. P(not red) = 1 − 3/8 = 5/8, which is the 5 yellow counters out of 8.

A fair die is rolled. Find the probability of a prime number.

  1. Prime numbers on the die: 2, 3 and 5, so 3 favourable faces.
  2. P = 3/6 = 1/2. (1 is not prime.)

Three coins are tossed. Find the probability of at least one head.

  1. The only outcome with no head is TTT, which is 1 of the 8 equally likely outcomes: P(no head) = 1/8.
  2. P(at least one head) = 1 − 1/8 = 7/8.

Your go. The first step is given. A card is picked at random from 20 cards numbered 1 to 20. Find the probability that it is a multiple of 5.

  1. There are 20 cards, each equally likely to be picked.
  2. Finish it, then check.
Check the answer

5, 10, 15 and 20: 4 cards out of 20, so P = 4/20 = 1/5, which is 0.2.

Check before you go on

1. A bag holds 3 red, 4 blue and 5 green counters. What is the probability of picking a blue counter?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A fair 10-sided die is numbered 1 to 10. Find the probability of a number greater than 7, as a decimal.

3. A ticket is drawn at random from tickets numbered 1 to 40. Find the probability that it is a multiple of 8, as a decimal.

Level 2 · Building

4. Two fair 4-sided dice (numbered 1 to 4) are rolled and the numbers added. Find the probability that the total is at least 6, as a decimal.

5. In a bag, P(red) = 0.15 and P(blue) = 0.4. All the other counters are green. Find P(green).

6. For which of these is “favourable outcomes ÷ all outcomes” a sensible way to find the probability?

Level 3 · Stretch

7. A bag holds 12 counters. 5 are red, and P(blue) = 1/4. All the other counters are green. How many green counters are there?

8. A bag holds 6 red counters and some blue ones. P(red) = 0.4. How many counters are in the bag altogether?

Something to think about

Two dice are rolled and the totals are 2, 3, …, 12. Which two totals have exactly the same probability as each other, and how can you tell from the grid without counting? Is there a total that has no partner?

One way to think about it

Totals the same distance either side of 7 match: 6 and 8, 5 and 9, 4 and 10, 3 and 11, 2 and 12. The grid is symmetrical about its diagonal of 7s. 7 has no partner: it is the middle total, with 6 cells.

Exit ticket

9. A card is picked from 10 cards numbered 1 to 10. Which event has the greatest probability?