Basic Rules of Probability Section 3.3   Probability Rules:   1.  P( ) = 0     *The smallest possible probability is of an impossible event (null set ).    2.  P(S) = 1    *The largest possible probability is of a certain event (event equal to S, the sample space   3.  0  P(E)  1        ** Probabilities exist between 0 and 1, inclusive.   **If you ever get a negative probability or a probability greater than 1, recheck you work and find your mistake.   Mutually exclusive events: Two events that cannot both occur at the same time. i.e. E  F =   if and only if E and F are mutually exclusive.   Example 1:  A  Pair of dice is rolled.  Notice that sample space is not =  {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} The sample space is all possible outcomes of two dice rolled.  Sample space has 36 elements. (see page 137)  Find the probability that:   a)  the sum is less than 5                                          b)     the sum is odd c)      the sum is less than 9 and odd. d)     The sum is less than 9 or odd   a)     P(a) = n(a)/n(S) = 6/36 = 1/6   b)     P(b) = 18/36 = ½   c)      P(c) = 12/36 = 1/3   d)     Use complement: the sum is greater than 9 and even = 4/36 = 1/9.  So           1 – 1/9 = 8/9         Notice all of the probabilities occur between 0 and 1.   More Probability Rules.   4.      P(E  F) = P(E) + P(F) – P(E  F)   5.      If E and F are mutually exclusive:          P(E  F) = P(E) + P(F)   6.      P(E) + P(E’) = 1               or                  1 – P(E) = P(E’)         or                  1 – P(E’) = P(E)   Example 2: Find the odds from EX 1.   a)     n(a) : n(a’) = 6 : 30  =  1 : 5   b)     n(b) : n(b’)  = 18 : 18  =  1 : 1   c)      n(c) : n(c’)  = 12 : 24  =  1 : 2   d)     n(d) : n(d’) =  32 : 4  = 8 : 1   #59 requires a lot of thought.  Use the Rules of Probability!   Back to Counting and Probability Main Page Back to the Survey of Math Ideas Home Page e-mail Questions and Suggestions