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Cambridge IGCSE0580

Introduction to probability

Mathematics 0580 Chapter Notes

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Introduction to probability
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In this chapter

  1. 1The probability scale
  2. 2Probability of a single event
  3. 3Probabilities from tables and Venn diagrams
  4. 4An event not happening
  5. 5Probability notation

1. The probability scale

The probability scale runs from 0 to 1. An impossible event has probability 0. A certain event has probability 1. Every other probability lies between them, so it can never be negative or greater than 1.

Give a probability as a fraction, decimal or percentage. A percentage must carry its percent sign.

Examiner insight

“Some gave the answer 24 indicating they did not know the basic property of probability has to be between 0 and 1.” “If candidates choose to give a percentage, they must include the percent-sign.”

Worked example 12 marks

The probability that a train is late is 7/20. Write this probability as a decimal and as a percentage.

  1. 1

    7/20 = 35/100 = 0.35 [1]

  2. 2

    0.35 = 35% [1]

2. Probability of a single event

To find the probability of a single event, count the outcomes that give the event and divide by the total number of possible outcomes.

When outcomes are equally likely, every one counts once. Check that the answer lies between 0 and 1.

P(event) = (number of outcomes that give the event)/(total number of possible outcomes)

Examiner insight

“Often questions like this give the probability as a fraction but candidates need to be able to be familiar with probabilities given as decimals or percentages as well.”

Worked example 13 marks

Two fair 4-sided dice have faces numbered 1 to 4. The dice are thrown, and the numbers are multiplied to give a total. The table shows all the possible results. Find the probability the total is

(i) 12

(ii) 5

(iii) a number that is less than 20.

dice A \ dice B1234
11234
22468
336912
4481216
  1. 1

    Total outcomes = 4 × 4 = 16. The total 12 appears twice, so(i) 2/16 = 1/8 [1]

  2. 2

    5 does not appear, so(ii) 0 [1]

  3. 3

    Every total is at most 16, so all 16 are less than 20:(iii) 16/16 = 1 [1]

3. Probabilities from tables and Venn diagrams

Tables, graphs and Venn diagrams give the counts. Read the numerator from the group the question names. Take the denominator from everyone in the whole group, not from the group named.

With two sets, the people in both sets are counted twice if you add the two set totals. Use the number who are in neither set to find the overlap.

in both = set 1 + set 2 - (total - neither)

Examiner insight

“The most common error was to select the wrong values, either finding the probability of selecting a right-handed person, or totalling some of the given numbers to make the numerator or denominator in their fraction.”

Worked example 11 mark

60 people complete a survey about hobbies. 41 people like reading books. 31 people like cookery. 11 do not like reading books and do not like cookery.

One of these 60 people is picked at random. Find the probability that they like cookery but they do not like reading books.

  1. 1

    Like at least one hobby = 60 - 11 = 49

  2. 2

    Like both = 41 + 31 - 49 = 23

  3. 3

    Cookery only = 31 - 23 = 8

  4. 4

    P = 8/60 = 2/15 [1]

4. An event not happening

The probability that an event does not occur is 1 minus the probability that it does. Subtract from 1, never from 100 unless the probability is a percentage.

Probabilities of all the possible outcomes add up to 1. Use this to find a missing probability.

P(not event) = 1 - P(event)

Examiner insight

“Many candidates did not understand that the calculation needed was 1 – 0.6 = 0.4 for the probability of picking a toy that was not wooden.” “The given probability of winning, 0.7, was often seen as the answer.”

Worked example 12 marks

A bag contains yellow counters, blue counters and red counters. Asif takes a counter at random from the bag. The probability that he takes a yellow counter is 27%. The probability that he takes a blue counter is 3/5. Find the probability that he takes a red counter.

  1. 1

    3/5 = 60%, so yellow or blue = 27% + 60% = 87% [1]

  2. 2

    Red = 100% - 87% = 13% (or 0.13 or 13/100) [1]

5. Probability notation

P(A) is the probability of A. P(A') is the probability of not A.

The two always add up to 1, so P(A') = 1 - P(A).

P(A') = 1 - P(A)

Worked example 12 marks

A counter is taken at random. B is the event that the counter is blue. P(B) = 0.8. Find P(B') and say what it means.

  1. 1

    P(B') = 1 - P(B) [1]

  2. 2

    = 1 - 0.8 = 0.2. This is the probability that the counter is not blue. [1]

Recap

  • The probability scale runs from 0 (impossible) to 1 (certain).
  • Give probabilities as fractions, decimals or percentages. A percentage needs its percent sign.
  • P(event) = outcomes that give the event ÷ total number of possible outcomes.
  • From a table or Venn diagram, the denominator is the whole group.
  • P(not event) = 1 − P(event). The probabilities of all outcomes add up to 1.
  • Extended: P(A) is the probability of A. P(A′) is the probability of not A.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. One of these cannot be a probability: 0.4, 7/5, 35%, 0. Write down which one and say why.2 marks
  2. A bag contains 5 red counters, 3 blue counters and 4 green counters. One counter is taken at random. Find the probability that it is blue.2 marks
  3. A fair 6-sided dice is thrown once. Find the probability of scoring more than 4.2 marks
  4. Do not use a calculator. The probability that it rains tomorrow is 0.35. Find the probability that it does not rain tomorrow.1 mark
  5. Do not use a calculator. The probability that a bus is late is 3/8. Find the probability that the bus is not late.2 marks
  6. 40 people are asked about drinks. 25 like tea. 18 like coffee. 7 like neither tea nor coffee. One of the 40 people is picked at random. Find the probability that this person likes both tea and coffee.3 marks
  7. A bag contains only red, white and green counters. The probability of taking a red counter is 0.35. The probability of taking a white counter is 0.4. Find the probability of taking a green counter.2 marks
  8. P(B') = 0.28. Find P(B) and say what P(B) means.2 marks
  9. Do not use a calculator. A ranger checks 80 birds at a lake. 46 have a leg ring. 30 have a wing tag. The number with neither is three times the number with both. One of the 80 birds is picked at random. A is the event that the bird has a leg ring but no wing tag. Find P(A'). Give your answer as a percentage.5 marks
  10. Do not use a calculator. A biased spinner lands on black, blue, red or green. It is spun once. The probability of black is 25%. The probability of blue is 2/5. The probability of red is twice the probability of green. R is the event that the spinner lands on red. Find P(R'). Give your answer as a fraction in its simplest form.5 marks

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