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All examples sourced from: Mathematical Studies SL, by Mal Coad et al., Haese & Harris Publications, 2nd Ed 2010



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The sample space of an event is a listing / picture or diagram of all the possible outcomes for that event (or experiment). The most common diagrams used to construct the sample space are: ›  Simply writing the outcomes in a list ›  Using a two-by-two table ›  Drawing a tree diagram ›  Drawing a venn diagram   Where it is often referred to as the universal set U



(i) List the sample space for (a) tossing a coin

(b) rolling a die

(ii) Use a tree diagram to write the sample space when: (a) tossing two coins (b) drawing 3 marbles from a bag with red and yellow marbles


Theoretical Probability is, sometimes called classical probability, is defined as:

number of times event occurs P(event occuring) = total number of possible outcomes n ( A) P ( A) = n (U )

›  Remember this is different to Experimental Probability because that is

based on a particular experiment or trial and the relative frequency calculated by that trial. ›  The difference can almost be described as theoretical being the “expected” probability and experimental as the “actual” probability. ›  In reality the experimental approaches the theoretical over time with many many trials.    

Hence the P(coin will land heads up) = ½ or P(choosing a diamond from a deck of cards) = ¼ Repeating an experiment one time or a hundred times has no effect on the “theoretical probability”, it remains the same.


The complementary probability of an event is the “NOT” case

P(event not occuring) = P ( A')

P ( A') = 1 − P ( A )

›  e.g if there is a 35% chance of rain tomorrow, there must be

a 65% chance that it will not rain.

Example 3   Find the probability that when rolling two dice they do not show doubles. ›  Rolling two die gives a total of 6 x 6 = 36 possible outcomes ›  Doubles are 1,1 and 2,2, … to 6,6 hence there are 6 outcomes. ›  P(not doubles) = 1 – P(doubles) = 1 – 6/36 = 30/36 = 5/6

Theoretical Probability