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Cambridge IGCSE0580

Conditional probability

Mathematics 0580 Chapter Notes

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Conditional probability
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Conditional probability notes

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In this chapter

  1. 1Given that: shrink the sample space
  2. 2Tree diagrams without replacement
  3. 3Just one: add the routes

1. Given that: shrink the sample space

A conditional probability question tells you something has already happened. Only the items that fit this fact are left to choose from.

The total is the number in the group you are told about, not the total of the whole table or Venn diagram. The top is the number in that group that also fits the question.

probability = (number that fit both conditions)/(number in the group you are told about)

Examiner insight

“The most common error involved an incorrect total number of students, usually 40 rather than the total number travelling by train.” “Many identified the correct 16 students but did not take into account that they were being chosen from just those wearing trainers.”

Worked example 13 marks

The table shows how 60 students travel to school. A student is chosen at random. Given that the student travels by bus, find the probability that the student is a girl.

buswalk
boys1416
girls1812
  1. 1

    students who travel by bus = 14 + 18 = 32 [1]

  2. 2

    girls who travel by bus = 18, so probability = 18/32 [1]

  3. 3

    = 9/16 [1]

2. Tree diagrams without replacement

When items are chosen and not replaced, the second probabilities depend on the first choice. The denominator is one less on the second branches.

Multiply along the branches to find the probability of a path. Two items picked together are picked without replacement, even when the question does not say so.

path probability = first branch × second branch

Examiner insight

“As two students were being picked then answers also needed to treat this as picking without replacement.” “Adding the relevant probabilities instead of multiplying was a common misunderstanding as was multiplying the product by 2.”

Worked example 13 marks

A bag holds 3 white counters and 5 black counters. Two counters are taken at random without replacement. Find the probability that both counters are white.

  1. 1

    first counter white: 3/8 [1]

  2. 2

    second counter white, 2 white left of 7 counters: 2/7 [1]

  3. 3

    3/8 × 2/7 = 6/56 = 3/28 [1]

3. Just one: add the routes

A question about "just one" of two items has two routes through the tree: A then B, and B then A. Work out each route by multiplying, then add the routes.

Choose from the group the question names. Do not use the whole sample space.

just one = (A then B) + (B then A)

Examiner insight

“Some of those went on to double the product but many did not and in a small number of cases candidates added the correct pair.”

Worked example 13 marks

The table shows information about the time that each of 90 trains is delayed. Two of the trains with delays of more than 30 minutes are chosen at random. Find the probability that just one of these trains is delayed for 45 minutes or less.

time (m minutes)0 < m ≤ 1010 < m ≤ 3030 < m ≤ 4545 < m ≤ 75
frequency1335366
  1. 1

    more than 30 minutes: 36 + 6 = 42 trains, and one is not replaced [1]

  2. 2

    either order: 2 × 36/42 × 6/41 [1]

  3. 3

    = 72/287 [1]

Recap

  • Extended: given that means the sample space shrinks to the group you are told about.
  • Extended: in a table or Venn diagram, divide by the total of that group.
  • Extended: two items taken together are taken without replacement, so the second denominator is one less.
  • Extended: multiply along the branches of a tree diagram.
  • Extended: for just one of two, add the two routes.

End-of-chapter exercise

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

  1. Do not use a calculator. A bag holds 5 red counters and 3 blue counters. One red counter is taken out and not replaced. A second counter is taken. Find the probability that it is red.2 marks
  2. Do not use a calculator. The table shows how 40 students get to a match. A student is chosen at random. Given that the student goes by coach, find the probability that the student is in Year 10. | | coach | car | | --- | --- | --- | | Year 10 | 9 | 11 | | Year 11 | 15 | 5 |3 marks
  3. Do not use a calculator. In a group of 40 people, 22 like tea, 15 like coffee and 6 like both. 9 like neither. One person who likes coffee is chosen at random. Find the probability that this person also likes tea.3 marks
  4. A box holds 4 red pens and 6 green pens. Two pens are taken at random without replacement. Find the probability that both are green.3 marks
  5. Do not use a calculator. A box holds 4 red pens and 6 green pens. Two pens are taken at random without replacement. Find the probability that exactly one pen is red.4 marks
  6. Do not use a calculator. A bag holds 5 mint sweets and 7 fruit sweets. Two sweets are taken at random without replacement. Find the probability that at least one sweet is mint.4 marks
  7. The table shows the language studied by 80 students. A student is chosen at random. Given that the student studies Spanish, find the probability that the student is in Year 11. | | French | Spanish | | --- | --- | --- | | Year 10 | 18 | 22 | | Year 11 | 25 | 15 |3 marks
  8. Do not use a calculator. 50 people were asked about sport. 28 swim, 20 cycle and 8 do both. Two people who swim are chosen at random. Find the probability that both of them also cycle.4 marks
  9. A drawer holds 8 black socks and 6 white socks. Two socks are taken at random without replacement. Given that the two socks are the same colour, find the probability that they are both black.5 marks
  10. A club has 60 members. 34 play chess, 27 play draughts and 8 play neither game. Two of the members who play chess are chosen at random. Find the probability that just one of them also plays draughts.5 marks

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