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 =

 

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!

 


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