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AI SL · 13.1

Probability

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13.1.7

Independent events

13.1.7

Independent events

Target: I can say whether two events are independent by comparing P(A ∩ B) with P(A) × P(B). (+5 more)

By the end of this section

Key points

Independent events

  • Two events are independent when one happening does not change the chance of the other: \(P(A|B)=P(A)\).
  • If A and B are independent, then \(P(A\cap B)=P(A)\times P(B)\).
  • To test for independence, work out \(P(A)\times P(B)\) and compare it with \(P(A\cap B)\). Equal means independent. Not equal means not independent.
  • Independent is not the same as mutually exclusive. If A and B cannot both happen, then \(P(A\cap B)=0\), so they are not independent unless one of them has probability 0.
  • Mutually exclusive: \(P(A\cap B)=0\). Independent: \(P(A\cap B)=P(A)\times P(B)\). Two events with non-zero probabilities cannot be both.

Using the rule

  • To find \(P(A\cap B)\): multiply \(P(A)\) by \(P(B)\). Only do this when the question tells you the events are independent, or you have shown it.
  • To find a missing probability, divide: \(P(B)=\frac{P(A\cap B)}{P(A)}\).
  • To find \(P(A\cup B)\) for independent events: \(P(A)+P(B)-P(A)P(B)\).

Sample space diagrams

  • For two dice, draw a grid: one die down the side, the other across the top. Each cell holds the score (a sum or a product).
  • Each cell is equally likely. \(P=\frac{\text{cells that fit}}{\text{all cells}}\).
  • Expected number of times = probability × number of trials. It need not be a whole number.

Venn diagrams from survey counts

  • Fill the overlap first, then work outwards. Finish with the people outside both circles: total − everyone inside.
  • A conditional probability shrinks the sample space. For \(P(A|B')\), only count people outside circle B, then ask how many of them are in A.
  • Write the number you divide by before you divide.

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: 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.
    Both happen, independent

    The probability that a student has a part-time job is 0.4. The probability that a student plays a team sport is 0.35. The two are independent. Find the probability that a student has a part-time job and plays a team sport.

    Working
    1. The rule

      \(P(A\cap B)=P(A)\times P(B)\)

      This is true only because the events are independent.

    2. Multiply

      \(0.4\times 0.35=0.14\)

      Give the answer as a decimal.

    Video walkthrough coming
    Testing for independence

    At a café, the probability that a customer buys coffee is 0.6 and the probability that a customer buys cake is 0.3. The probability that a customer buys both is 0.2. Are buying coffee and buying cake independent?

    Working
    1. The product

      \(P(\text{coffee})\times P(\text{cake})=0.6\times 0.3=0.18\)

      This is what P(both) would be if the events were independent.

    2. Compare

      \(P(\text{both})=0.2\ne 0.18\)

      The two are not equal.

    3. Conclude

      Not independent

      Customers who buy coffee are more likely to buy cake. A conclusion needs the comparison, not just the word.

    Video walkthrough coming
    A missing probability

    Whether the school bus is late and whether it rains are independent. The probability that the bus is late is 0.25. The probability that the bus is late and it rains is 0.07. Find the probability that it rains.

    Working
    1. The rule

      \(0.07=0.25\times P(\text{rain})\)

      Independent, so the product gives P(both).

    2. Divide

      \(P(\text{rain})=0.07\div 0.25=0.28\)

      Check: 0.25 × 0.28 = 0.07.

    Video walkthrough coming
    Product of two dice

    Two fair six-sided dice are rolled and the score is the product of the two numbers. The game is played 108 times. Use a sample space diagram to find how many times you would expect a score of 5 or less.

    ×1234561123456224681012336912151844812162024551015202530661218243036
    Working
    1. The diagram

      6 by 6 grid of products (shown)

      First die down the side, second across the top. 36 equally likely cells.

    2. Count

      Scores of 5 or less: 10 cells

      Look along the top-left corner of the grid.

    3. Probability

      \(P=\frac{10}{36}=\frac{5}{18}\)

      Cancel the fraction if you can.

    4. Expected number

      \(\frac{5}{18}\times 108=30\)

      Probability × number of trials.

    Video walkthrough coming
    A four-sided and an eight-sided die

    A fair four-sided die (numbered 1 to 4) and a fair eight-sided die (numbered 1 to 8) are rolled. The two scores are added. Find the probability that the total is a prime number.

    +12345678123456789234567891034567891011456789101112
    Working
    1. The diagram

      4 rows by 8 columns of totals (shown)

      4 × 8 = 32 equally likely outcomes.

    2. Prime totals

      Totals 2, 3, 5, 7 and 11 are prime: 13 cells

      1 is not prime. The largest total is 12.

    3. Probability

      \(P=\frac{13}{32}\)

      Cancel to lowest terms.

    Video walkthrough coming
    A survey in a Venn diagram

    In a survey of 140 students, 56 take drama, 72 take music and 20 take both. (a) Draw a Venn diagram. (b) Find the probability that a student takes neither. (c) Given that a student does not take music, find the probability that they take drama.

    36205232DramaMusic
    Working
    1. Middle first

      Both: 20

      The overlap is given.

    2. Only one

      Drama only: 56 − 20 = 36. Music only: 72 − 20 = 52

      Take the overlap out of each circle.

    3. Neither

      140 − (36 + 20 + 52) = 32

      Total minus everyone inside the circles. Part (b): \(P=\frac{32}{140}=\frac{8}{35}\)

    4. Given not music

      Not music: 36 + 32 = 68 students. \(P=\frac{36}{68}=\frac{9}{17}\)

      The sample space is now the 68 students outside the music circle.

    Video walkthrough coming
    Mutually exclusive is not independent

    Events A and B are mutually exclusive, with P(A) = 0.25 and P(B) = 0.4. (a) Find P(A) × P(B). (b) Are A and B independent? Give a reason.

    Working
    1. Mutually exclusive

      \(P(A\cap B)=0\)

      They cannot both happen.

    2. The product

      \(P(A)\times P(B)=0.25\times0.4=0.1\)

      Independent events need P(A ∩ B) to equal this product.

    3. Compare

      \(0\neq0.1\), so A and B are not independent.

      If one of two possible events happens, the other cannot. Knowing that changes its probability, so they are dependent. Events with non-zero probabilities are never both mutually exclusive and independent.

    Video walkthrough coming