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13.6 More Tree Diagrams

GCSE Maths Foundation · Edexcel · Chapter 13: Probability

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: independent and dependent events, choosing without replacement, and “at least one”. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Decide whether two events are independent or dependent
  • Draw tree diagrams for choosing with and without replacement
  • Use tree diagrams where the second probabilities depend on the first outcome
  • Use 1 − P(none) to find the probability of “at least one”

Key words

  • Independent: The first outcome does not change the probabilities for the second.
  • Dependent: The first outcome does change the probabilities for the second.
  • With replacement: The item is put back, so the second pick has the same probabilities as the first.
  • Without replacement: The item is not put back, so there is one fewer item for the second pick.

Independent or dependent?

Ask: “Does what happened first change the chances for the second event?” If not, the events are independent. If it does, they are dependent.

  • Flipping a coin, then rolling a dice; Type Independent
  • Taking a counter, putting it back, then taking another; Type Independent
  • Taking a sweet, eating it, then taking another; Type Dependent
  • Whether it rains, and whether a train is late; Type Dependent

With and without replacement

A bag holds 5 green and 3 yellow sweets. Two sweets are taken at random.

Worked example: with replacement

The first sweet is put back, so the bag still has 8 sweets for the second pick.
P(both yellow) = 3/8 × 3/8 = 9/64.

Answer: 9/64

Worked example: without replacement

The first sweet is eaten. If it was yellow, the bag now has 7 sweets and only 2 are yellow.
P(both yellow) = 3/8 × 2/7 = 6/56 = 3/28.

Answer: 3/28

Key fact: Without replacement: the total goes down by 1, and the number of the colour that was taken goes down by 1.

Tree diagrams without replacement

Worked example: using the tree

P(both the same colour) = P(GG) + P(YY) = 20/56 + 6/56 = 26/56 = 13/28.
P(one of each colour) = P(GY) + P(YG) = 15/56 + 15/56 = 30/56 = 15/28.
Check: 13/28 + 15/28 = 1.

Answer: 13/28 and 15/28

When the second event depends on the first

Dependent events are not only about bags of sweets. Here the chance of the train being late depends on the weather.

Worked example: probability the train is late

Late can happen two ways: rain and late, or no rain and late.
P(late) = 0.12 + 0.07 = 0.19.

Answer: 0.19

“At least one” and three events

“At least one” means one, two, three … – lots of paths. It is quicker to use the opposite: none at all.

Key fact: P(at least one) = 1 − P(none)

Worked example: three sets of traffic lights

At each of three sets of lights, P(red) = 0.4, independently. Find P(at least one red).
P(no red) = 0.6 × 0.6 × 0.6 = 0.216.
P(at least one red) = 1 − 0.216 = 0.784.

Answer: 0.784

Common mistakes

  • Not changing the second-stage probabilities when items are not replaced.
  • Reducing the top number for the wrong colour: only the colour that was taken goes down.
  • Adding along branches instead of multiplying.
  • Working out every path for “at least one” and missing some. Use 1 − P(none).

Revision questions

Try these, then check your answers below each one.

Question 1. Which pair of events is dependent? (Choose from: Flipping a coin twice, Rolling a dice and spinning a spinner, Taking two sweets from a bag and eating the first, Taking a counter, replacing it, then taking another.)

Answer: Taking two sweets from a bag and eating the first

Working: Eating the first sweet changes what is left in the bag, so the second probability depends on the first.

Question 2. A bag holds 3 red and 5 blue counters. Two are taken at random without replacement. Work out P(both blue). Give a fraction in its simplest form.

Answer: 5/14

Working: 5/8 × 4/7 = 20/56 = 5/14.

Question 3. A bag holds 3 red and 5 blue counters. A counter is taken, put back, and a second counter is taken. Work out P(both red). Give a fraction.

Answer: 9/64

Working: 3/8 × 3/8 = 9/64.

Related lessons

Continue in the full course

This lesson is section 13.6 (EGCMF1306) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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