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AI SL · 13.1

Probability

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13.1.3

Tree diagrams and sample spaces

13.1.3

Tree diagrams and sample spaces

Target: I can draw a tree diagram, with or without replacement, and find a probability by multiplying along branches and adding. (+4 more)

By the end of this section

Key points

Tree diagrams

  • Use a tree when one event follows another. Each set of branches from one point adds to 1.
  • Multiply along the branches to get the probability of one path ("and"). Add the paths that match the question ("or").
  • Without replacement: the second stage changes. If 9 items are in the bag, the second draw is from 8.
  • "At least one" is often quicker as 1 − P(none).
  • Conditional probability here is read straight from the tree: the second-stage branch is the probability given the first. You do not need Bayes' theorem.
  • With replacement (or independent repeated trials) every stage has the same branches. "Exactly two of three" is a set of paths: list them, multiply along each, then add.

Sample space diagrams

  • A sample space diagram lists every equally likely outcome of two events, in a grid or a list.
  • Write the outcomes in order, row by row, so none is missed or repeated. BG and GB are different.
  • P = favourable cells ÷ all cells. Read the question carefully: "8 or more" includes 8.

Venn diagrams with probabilities

  • Fill the overlap first, then the "only" regions by subtraction. The four regions add to 1.
  • \(P(\text{neither})=1-P(A)-P(B)+P(A\cap B)\): the overlap is counted once, not twice.

Explore: the sample space of two dice

Each of the 36 cells is one equally likely outcome. Choose an event, or press cells to make your own. Add a second event to see "and", "or" and "given".

    Cells in A, n(A)
    Cells in B, n(B)
    Cells in both, n(A ∩ B)
    Favourable cells

    Choose an event to see its cells.

    1. Choose "the sum is 7". Which cells are lit, and why is that line longer than the line for a sum of 12?
    2. Choose A: the sum is at least 10 and B: the first die is greater than the second, then "A given B". Why is the count divided by n(B) and not by 36?
    3. Choose "A or B". Check that n(A ∪ B) = n(A) + n(B) − n(A ∩ B) with the counts shown.
    4. Press cells to make an event with probability 1/4. How many cells do you need? Is there only one way to do it?

    Explore: a two-stage tree diagram

    Fill the tree from a bag of counters, with or without replacement, or type your own probabilities in the orange boxes. The grey box on each pair is 1 minus its partner.

    The tree comes from

    The four end products

    Read out

    Total of the four end products

    Choose a bag or type your own probabilities.

    1. Choose "without replacement". Compare the second-stage branches after a red and after a blue. Why are they different? Then choose "with replacement": what changes?
    2. Look at the four end products. Why must they add to 1?
    3. Which is bigger, P(exactly one) or P(at least one)? Can they ever be equal? Try the counts to see.
    4. For "first, given second", why is the bottom of the fraction the probability of the second counter and not the first branch?

    Explore: two events in a Venn diagram

    Type P(A), P(B) and P(A ∩ B). The four regions fill in. Choose what to shade and read the working. The circles are not drawn to scale.

    Shade

    P(A ∪ B)
    P(A′)
    P(A | B)
    P(B | A)
    P(A) × P(B)

    Type three probabilities to fill the diagram.

    1. Start with P(A) = 0.5 and P(B) = 0.4. Change only P(A ∩ B) until the events are independent. What value does it need, and how do you know?
    2. Choose "Mutually exclusive". Look at P(A | B). Why is it 0? Are the events independent?
    3. Compare P(A | B) with P(B | A). When are they the same, and why is it usually not so?
    4. Try a P(A ∩ B) bigger than P(A). What does the sentence say, and why can it never happen?

    Worked examples

    Problems like the ones in class. Pick one, try it, then open the working.
    A tree with conditional branches

    Maya cycles to school with probability 3/5, and otherwise walks. If she cycles, the probability that she arrives early is 3/4. If she walks, the probability that she arrives early is 1/3. Find the probability that she arrives early. Give your answer as a decimal to 3 significant figures.

    Working
    1. 3/5Cycle3/4Early9/201/4Late3/202/5Walk1/3Early2/152/3Late4/15
      Draw the tree

      First stage: cycle or walk. Second stage: early or not early.

      The second branches depend on the first, so each pair of branches has its own probabilities.

    2. Multiply along the branches

      \(P(\text{cycle, early})=\frac35\times\frac34=\frac{9}{20}\) \(P(\text{walk, early})=\frac25\times\frac13=\frac{2}{15}\)

      "And" means multiply along a path.

    3. Add the paths that give early

      \(P(\text{early})=\frac{9}{20}+\frac{2}{15}=\frac{7}{12}\) = 0.583

      "Or" means add the end outcomes that match.

    Video walkthrough coming
    Without replacement

    A bag holds 5 green counters and 4 white counters. Two counters are taken at random, one after the other, without replacement. Find the probability that they are the same colour. Give your answer as a decimal to 3 significant figures.

    Working
    1. 5/9G1/2G5/181/2W5/184/9W5/8G5/183/8W1/6
      The tree

      After a green first, the bag has 4 green and 4 white (8 counters).

      Taking a counter out changes the second-stage probabilities.

    2. Same colour: GG or WW

      \(P(GG)=\frac59\times\frac48=\frac{5}{18}\) \(P(WW)=\frac49\times\frac38=\frac{1}{6}\)

      Multiply along each path.

    3. Add

      \(\frac{5}{18}+\frac{1}{6}=\frac{4}{9}\) = 0.444

      The two paths cannot both happen, so add.

    Video walkthrough coming
    A Venn diagram with probabilities

    In a café, the probability that a customer buys a coffee is 0.55 and the probability that a customer buys cake is 0.4. The probability that a customer buys both is 0.15. Find the probability that a customer buys neither.

    Working
    1. Overlap first

      P(both) = 0.15

      Same method as with counts, but the regions add to 1.

    2. Only coffee, only cake

      Coffee only = 0.55 − 0.15 = 0.4. Cake only = 0.4 − 0.15 = 0.25

      Take the overlap away from each total.

    3. 0.40.250.150.2FK
      Neither

      1 − 0.4 − 0.15 − 0.25 = 0.2

      The four regions add to 1.

    Video walkthrough coming
    A sample space diagram of sums

    A fair six-sided die and a fair four-sided die are rolled and the two scores are added. Find the probability that the sum is 8 or more.

    Working
    1. +12341234523456345674567856789678910
      Draw the grid

      6 rows for the six-sided die, 4 columns for the four-sided die: 24 equally likely cells.

      Each cell is the sum of its row score and its column score.

    2. Count

      Cells with sum 8 or more: 6

      Check each row from the bottom. 8 counts.

    3. Probability

      \(P=\frac{6}{24}=\frac{1}{4}\) = 0.25

      Favourable cells over all cells.

    Video walkthrough coming
    A sample space for two children

    A family has two children. Each child is equally likely to be a boy (B) or a girl (G). Find the probability that at least one child is a girl. Give your answer as a decimal.

    Working
    1. BGBBBBGGGBGG
      List all outcomes

      BB, BG, GB, GG

      The order matters: BG and GB are different outcomes. Four equally likely outcomes.

    2. Count

      At least one G: BG, GB, GG. That is 3 outcomes.

      Only BB has no girl.

    3. Probability

      \(P=\frac34\) = 0.75

      3 out of 4.

    Video walkthrough coming
    Listing arrangements

    The letters N, E and T are written on three cards, which are put in a row in a random order. Find the probability that the two consonants are next to each other. Give your answer as a decimal to 3 significant figures.

    Working
    1. List every arrangement

      NET, NTE, ENT, ETN, TNE, TEN

      Work systematically: first letter N, then E, then T. There are 6.

    2. Mark the favourable ones

      Consonants together (E at an end): NTE, ENT, ETN, TNE. That is 4.

      N and T are the consonants. They touch when E is first or last.

    3. Probability

      \(P=\frac{4}{6}=\frac{2}{3}\) = 0.667

      Give the fraction, or 0.667 as a decimal.

    Video walkthrough coming
    Repeated trials: a tree with three stages

    A basketball player scores each free throw with probability 0.7, independently of the other throws. She takes three free throws. Find the probability that she scores exactly two.

    Working
    1. 0.7S0.7S0.7S0.3430.3M0.1470.3M0.7S0.1470.3M0.0630.3M0.7S0.7S0.1470.3M0.0630.3M0.7S0.0630.3M0.027
      The tree

      Each throw: S (score) with 0.7, M (miss) with 0.3. Three stages.

      The throws are independent, so every stage has the same branches. The probabilities do not change.

    2. Find the paths with exactly two scores

      SSM, SMS, MSS

      Two scores and one miss, in any order. List them so none is missed.

    3. Multiply along each path

      \(0.7\times0.7\times0.3=0.147\) for each path

      Each path has the same probability: two 0.7 and one 0.3.

    4. Add the three paths

      \(P=3\times0.147=0.441\)

      The paths cannot happen together, so add.

    Video walkthrough coming