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Cambridge IGCSE0607

Introduction to probability

International Mathematics 0607 Chapter Notes

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Introduction to probability
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In this chapter

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

1. The probability scale

A probability is a number on a scale from 0 to 1. It cannot be greater than 1, so an answer such as 11 or 7/5 is not a probability.

Give a probability as a fraction, a decimal or a percentage. Never write it as a ratio or with the words ‘out of’.

Examiner insight

“Candidates should be aware that probabilities cannot be greater than 1 and so an answer of 11 is impossible.” “A few gave answers greater than 1, which they should know is not possible for a probability.”

Worked example 11 mark

Samir rolls a fair six-sided die numbered 1 to 6. The probability of event A is 5/6. Tick the one statement that could describe event A.

The number rolled is less than 5.

The number rolled is prime.

The number rolled is greater than 1.

The number rolled is even.

  1. 1

    less than 5 is 1, 2, 3, 4, so 4/6

  2. 2

    prime is 2, 3, 5, so 3/6. Even is 2, 4, 6, so 3/6

  3. 3

    greater than 1 is 2, 3, 4, 5, 6, which is 5 of the 6 numbers, so 5/6: tick this one [1]

2. The probability of a single event

To find the probability of a single event, count the outcomes that give the event. Divide by the total number of possible outcomes. The answer needs both the numerator and the correct denominator.

On a fair 6-sided die, each face has the same probability of 1/6. The probability of rolling a 2 is 1/6, not 2/6.

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

Examiner insight

“Candidates wrote the score required, 2, over the total number,6.” “Of those who got this part wrong, some wrote the correct numerator with no denominator and others wrote the correct numerator with an incorrect denominator.”

Worked example 12 marks

The table shows the numbers of letters delivered to each of 90 houses one day.

(a) Find the value of n.

(e) One of the 90 houses is selected at random. Find the probability that the house had more than 4 letters delivered.

number of letters0123456
frequency272315108n3
  1. 1

    the frequencies add to 90: n = 90 - (27 + 23 + 15 + 10 + 8 + 3) = 90 - 86 = 4 [1]

  2. 2

    more than 4 letters means 5 or 6 letters: 4 + 3 = 7 houses

  3. 3

    probability = 7/90 [1]

3. Probability from tables and Venn diagrams

Problems may give the information in a table, a graph or a Venn diagram (limited to two sets in the Core course). Read off the number you need. Then find the total.

For a table, the total is the sum of the frequencies. ‘More than 4’ does not include 4. If two groups fit the event, add their frequencies: in the letters question above, giving only the last frequency loses the mark.

Examiner insight

“A small number of candidates gave their answer as 3/90.” “A few did not use the value 25 as the denominator.”

Worked example 11 mark

The table shows the marks scored by 50 students in a test. One of these students is chosen at random. Find the probability that this student scored more than 25 marks.

mark1 to 56 to 1011 to 1516 to 2021 to 2526 to 3031 to 35
frequency137129117
  1. 1

    more than 25 marks: the groups 26 to 30 and 31 to 35, so 11 + 7 = 18 students

  2. 2

    probability = 18/50 (= 9/25) [1]

4. An event not happening

An event either happens or it does not. So the probability that it does not occur is 1 minus the probability that it does. If the probability that a counter is blue is 0.8, the probability that it is not blue is 1 - 0.8 = 0.2.

To find the last probability, add the known ones and subtract the sum from 1.

P(not happening) = 1 - P(happening)

sum of all the probabilities = 1

Examiner insight

“This part was not particularly well answered as many candidates found the sum of the given two fractions but then did not subtract this from 1 to find the required probability.” “Some candidates who realised that the sum of the probabilities equals one, did not gain the mark because of their inability to process the decimal calculations correctly.”

Worked example 11 mark

The probability that any train arrives at station A on time is 5/6. Write down the probability that a train arriving at station A is not on time.

  1. 1

    P(not on time) = 1 - P(on time)

  2. 2

    = 1 - 5/6 = 1/6 [1]

5. Probability notation

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

The rule for an event not happening is then short. For example, if P(B) = 0.8, then P(B') = 1 - 0.8 = 0.2.

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

Worked example 12 marks

The probability that a student walks to school is P(W) = 0.35. Find P(W') and say what it means.

  1. 1

    P(W') is the probability that a student does not walk to school [1]

  2. 2

    P(W') = 1 - 0.35 = 0.65 [1]

Recap

  • The probability scale runs from 0 to 1. A probability is never greater than 1.
  • Give a probability as a fraction, a decimal or a percentage. Never a ratio, never ‘out of’.
  • Probability = outcomes that give the event ÷ total number of possible outcomes.
  • From a table, add the frequencies for the total. ‘More than 4’ does not include 4.
  • P(not happening) = 1 − P(happening). All the probabilities add up to 1.
  • Extended: P(A) is the probability of A. P(A′) is the probability of not A, and P(A′) = 1 − P(A).

End-of-chapter exercise

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

  1. Do not use a calculator. Which of these cannot be a probability? 0.3, 7/5, 40%, 11 mark
  2. Do not use a calculator. A fair 6-sided die is rolled. Find the probability of rolling a 2.1 mark
  3. Do not use a calculator. A bag holds 4 red counters, 5 blue counters and 3 green counters. One counter is chosen at random. Find the probability that it is green. Give your answer as a fraction in its simplest form.2 marks
  4. A group of 20 students were asked how many pets they have. One of the students is chosen at random. Find the probability that this student has at least 2 pets. Give your answer as a decimal. | number of pets | 0 | 1 | 2 | 3 | | --- | --- | --- | --- | --- | | number of students | 6 | 9 | 3 | 2 |2 marks
  5. Do not use a calculator. The probability that a bus is late is 15%. Find the probability that the bus is not late. Give your answer as a percentage.1 mark
  6. Do not use a calculator. In a game, the probability of winning is 2/5 and the probability of a draw is 1/4. Find the probability of losing.3 marks
  7. Do not use a calculator. The probability that it rains tomorrow is P(R) = 0.37. Find P(R').1 mark
  8. A spinner has only red, blue and green sectors. P(red) = 0.3 and P(blue) = 0.45. (a) Find P(green). (b) Find P(blue').3 marks
  9. A repair shop tests 48 phones. 29 have a cracked screen (C), 23 have a weak battery (B) and 7 have neither fault. One of the 48 phones is chosen at random. (a) Find the probability that the phone has both faults. (b) Find P(C'). (c) Find the probability that the phone has exactly one of the two faults. Give your answer as a fraction in its simplest form.6 marks
  10. A car park holds only electric (E), diesel (D) and hybrid (H) cars. One car is chosen at random. P(E) = 0.15 and P(D') = 0.6. There are 27 hybrid cars. (a) Find the number of cars in the car park. (b) 12 more hybrid cars arrive and no cars leave. One car is now chosen at random. Find P(E'). Give your answer as a fraction in its simplest form.7 marks

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