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