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Probability Trees

discia.org 10 questions · 37 marks
Name Date
  1. Either paper 3 marks

    A bag contains 2 blue tiles and 7 grey tiles.

    A tile is taken from the bag at random, its colour is written down, and it is put back in the bag.

    A second tile is then taken from the bag at random.

    The tree diagram is not complete.

    BlueGreyBlueGreyBlueGrey2979........................................
    (a)

    State the probability that the second tile taken is grey.

    (b)

    Find the probability that both tiles taken are blue.

    Show answer
    (a) \(\frac{7}{9}\)
    (b) \(\frac{4}{81}\)
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  2. Either paper 3 marks

    The probability that the train is late on Tuesday is \(0.4\).

    The probability that it is late on Wednesday is \(0.1\).

    The tree diagram shows this information. One probability is missing.

    LateOn timeLateOn timeLateOn time0.4?0.10.90.10.9
    (a)

    State the probability that should be written on the branch marked with a question mark.

    (b)

    Find the probability that the train is on time on both days.

    Show answer
    (a) \(0.6\)
    (b) \(0.54\)
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  3. Either paper 3 marks

    Owen and Leila each take a piano exam.

    The probability that Owen passes is \(0.7\).

    The probability that Leila passes is \(0.3\).

    The two results are independent. The tree diagram shows Owen's result, then Leila's result.

    PassFailPassFailPassFail0.7..........0.3..............................
    (a)

    Complete the tree diagram.

    (b)

    Work out the probability that Owen fails and Leila passes.

    Show answer
    (a) Owen fails \(0.3\). Leila under Owen passing: fails \(0.7\). Leila under Owen failing: passes \(0.3\), fails \(0.7\).
    (b) \(0.09\)
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  4. Either paper 3 marks

    A fair, ordinary dice is rolled and a counter is taken at random from a bag.

    The tree diagram shows the probabilities.

    More than 44 or lesswhitebluegreenwhitebluegreen1323220920920220920920
    (a)

    How do the probabilities show that all the counters in the bag are white, blue or green?

    (b)

    State the probability that the counter is white or green.

    (c)

    State the probability that the dice lands on a number greater than 4 and the counter is white.

    Show answer
    (a) The three probabilities from each counter node add up to 1: \(\frac{ 2 }{ 20 } + \frac{ 9 }{ 20 } + \frac{ 9 }{ 20 } = 1\), so no other colour is possible.
    (b) \(\frac{11}{20}\)
    (c) \(\frac{1}{30}\)
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  5. Either paper 4 marks

    Leila plays one game of darts and one game of snooker.

    The probability that she wins the game of darts is \(0.8\).

    The probability that she wins the game of snooker is \(0.2\).

    The two results are independent.

    WinsLosesWinsLosesWinsLosesWins, WinsWins, LosesLoses, WinsLoses, LosesOutcome0.80.20.20.80.20.8
    (a)

    Find the probability that Leila wins both games.

    (b)

    Find the probability that Leila wins neither game.

    Show answer
    (a) \(0.16\)
    (b) \(0.16\)
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  6. Either paper 3 marks

    On any Monday, the probability that Amara is late for school is \(\frac{ 5 }{ 9 }\).

    On the same Monday, the probability that Ben is late for school is \(\frac{ 5 }{ 6 }\).

    Whether one of them is late does not affect the other.

    Amara lateAmara ontimeBen lateBen on timeBen lateBen on time594956165616

    Work out the probability that exactly one of Amara and Ben is late.

    Show answer
    \(\frac{25}{54}\)
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  7. Either paper 5 marks

    Isla takes three shots at a netball goal.

    The probability that she scores with any one shot is \(0.4\), and the shots are independent of each other.

    ScoresMissesScoresMissesScoresMissesScoresMissesScoresMissesScoresMissesScoresMisses0.40.60.40.60.40.60.40.60.40.60.40.60.40.6
    (a)

    Work out the probability that Isla scores with all three shots.

    (b)

    Work out the probability that Isla scores with at least one of the three shots.

    Show answer
    (a) \(0.064\)
    (b) \(0.784\)
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  8. Either paper 4 marks

    There are 40 girls and 80 boys in a choir.

    If a girl is chosen at random, the probability that she plays a musical instrument is 0.3.

    If a boy is chosen at random, the probability that he plays a musical instrument is 0.49.

    One of the 120 people is chosen at random.

    Show that the probability that this person plays a musical instrument is more than 0.4.

    Show answer
    \(\frac{ 51.2 }{ 120 } \approx 0.427\), which is more than 0.4
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  9. Either paper 3 marks

    Leila plays two games of Scrabble against Owen.

    In each game, Leila could win, draw or lose.

    In each game, the probability that Leila wins is \(0.22\).

    In each game, the probability that Leila draws is \(0.15\).

    Calculate the probability that Leila wins exactly one of the two games.

    Show answer
    \(0.3432\)
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  10. Either paper 6 marks

    Rosa plays a game with an ordinary, fair dice.

    If she rolls a 1 or a 2 she wins.

    If she rolls a 3 or a 4 she loses.

    If she rolls any other number she rolls again.

    When she has to roll again, she wins if she rolls a 6. Otherwise she loses.

    Draw a tree diagram for the game and write the probability on each branch.

    Is Rosa more likely to win or to lose?

    You must work out the probability that Rosa wins.

    Show answer
    \(P(\text{win}) = \frac{7}{18}\), which is less than \(\frac{1}{2}\), so Rosa is more likely to lose
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discia.org Probability Trees · sheet 1
Answers Probability Trees · sheet 1
  1. (a) \(\frac{7}{9}\)
    (b) \(\frac{4}{81}\)
  2. (a) \(0.6\)
    (b) \(0.54\)
  3. (a) Owen fails \(0.3\). Leila under Owen passing: fails \(0.7\). Leila under Owen failing: passes \(0.3\), fails \(0.7\).
    (b) \(0.09\)
  4. (a) The three probabilities from each counter node add up to 1: \(\frac{ 2 }{ 20 } + \frac{ 9 }{ 20 } + \frac{ 9 }{ 20 } = 1\), so no other colour is possible.
    (b) \(\frac{11}{20}\)
    (c) \(\frac{1}{30}\)
  5. (a) \(0.16\)
    (b) \(0.16\)
  6. \(\frac{25}{54}\)
  7. (a) \(0.064\)
    (b) \(0.784\)
  8. \(\frac{ 51.2 }{ 120 } \approx 0.427\), which is more than 0.4
  9. \(0.3432\)
  10. \(P(\text{win}) = \frac{7}{18}\), which is less than \(\frac{1}{2}\), so Rosa is more likely to lose
discia.org Probability Trees · sheet 1