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Tree diagrams


Theoretical Probability: P(E) of any event is expressed as a fraction:



We can find this probability by two methods; the first is the tree diagram


A tree diagram lists all the possible outcomes and we circle (or color) all favorable outcomes.


Example: Suppose I have a coin and a six-sided dice:  On the coin, I can get a heads or a tail and on the dice I could get a 1,2,3,4,5 or 6. Lets say I want to know the probability of getting a head and an odd number if I tossed both the die and the coin.   So I would make a tree diagram and it would look like this:


Head   Tail       Head   Tail     Head   Tail     Head   Tail     Head   Tail    Head    Tail

    1        1             2        2          3         3          4         4           5        5          6         6


Notice there are twelve possible outcomes (count up all the numbers on the last row) and only three favorable outcomes (count up the red on the bottom row).  I would write this probability as P(head,odd)=                       I can reduce this to P(head,odd) =


Example 2: A Deli offers four different types of meats (Ham, Pastrami, Roast Beef, Turkey), two types of cheeses (Swiss, American, Mozzerella) and three types of breads White (W), Rye (R), Bran (B)).  They place one of each type of sandwich in a large bag.  What is the probability if you picked a sandwich randomly that you would gets a Ham and Swiss on either rye or bran bread?


               Ham                             Pastrami                           Roast Beef                         Turkey

      Swiss     American         Swiss       American          Swiss      American           Swiss       American   

W  R    B       W  R   B       W  R    B     W  R   B        W  R    B    W  R   B       W  R    B      W  R   B


Remember count only the bottom row for all possible outcomes (there are 24 outcomes)  and only two are favorable (Ham with Swiss on Rye or Bran)

So P(Ham, Swiss on Rye or Bran) =