Venn Diagrams and Tree Diagrams

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Conditional Probability, part 1 128-1.8.a
9:51 — by HCCMathHelp

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Key Questions

  • A Venn diagram is graphical device that depicts sample spaces and random events symbolically ( and that are named after their originator, John Venn). Definition from Statistics for Business and Economics Book, Third Edition, by Heinz Kohler
    Venn diagrams which are usually used in set theory can also be used to solve some probability problems. For example, the diagram shows how children come to school by walking (W), by bicycle (B) or by car (C).
    enter image source here
    Using the information on the Venn diagram to find the probability that a child picked at random, then Possibility space = 6 + 5 + 7 + 10 + 9 + 8 + 12 = 57 and probability of children who walk = 7 + 8 + 5 + 9 = 29P (W) = 29/57, or who use a car = 6 + 5 + 10 + 9 = 30P (C) = 30/57 = 10/19
    Read more here !

  • It helps to visualise conditional probability problems.

    For example, if you toss a coin twice, and you would like to work out the probability of obtaining 2 heads in a row. You would tree diagram a simple plot like this with the relevant probabilities: One thing to not is the sum of the probabilities from each 'node' of the branch should always add to precisely 1 - in this case the chances of heads and tails are equal so you have a 0.5 chance of obtaining either.

    enter image source here

    To find the probability we find all the branches which lead to the desired outcome, in this case there is just one, and we multiply the probabilities together. So the probability of obtaining to heads is 0.5 x 0.5 = 0.25

    If we wanted to workout the probability of obtaining a heads and then tails in any particular order, you would look at both branches which lead to this outcome. In this case there are 2:

    enter image source here

    Work out the relevant probability from both branches, in this case the chance of going down each branch is 0.5 x 0.5 = 0.25. We then add these probabilities together (0.25 + 0.25) to obtain a probability of 0.5 of obtaining a heads and a tails, or a tails then heads. Makes sense when you think about it!

    That covers the basics, you can add a further branch to work out probabilities of different outcomes from more than 2 coin tosses, and use it to model more complex events with differing probabilities.

  • They can make you see whether you can ADD probabilities or have to multiply.

    Let's take the tree first. You throw three coins. What are the chances there are at least two heads? Draw the tree!
    First split: H 0.5 -- T 0.5 let's do the H to the left.
    Second splits: again H 0.5 -- T 0.5
    So H-H has a chance of 0.5 times 0.5 = 0.25 and we can stop left-left
    T-T is already a looser (0.25)
    We still have H-T and T-H each with a chance of 0.25
    Third throw: they both have a chance of 0.5 so:
    H-T-H = 0.5 x 0.25 = 0.125
    T-H-H = 0.5 x 0.25 = 0.125
    Adding up, we have 0.25 + 0.125 + 0.125 =0.5
    If we had wanted to know the chance of exactly two heads we would have had to go on with the left-left, because H-H-H would then be excluded.
    Summary: down a tree, you multiply -- at the ends you add.

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