9.7 Probability

9.7 Probability

 

Definitions

An experiment is an activity under consideration.
ie: tossing a coin or rolling a die.

Each possible observation in an experiment is an outcome.

The set of possible outcomes of an experiment is called sample space / outcome set.
ie: the sample set when rolling a die is

Any subset of a sample space is called an event.

Multiple Experiments

If the events are independent, the outcome of one does not influence the other.
ie: rolling a die does not effect the outcome of flipping a coin

Events are mutually exclusive if they cannot both occur.
ie: -when rolling a die one time, rolling an even and odd number are mutually exclusive events
     -When picking a marble from a bag with three blue marbles and three red marbles, picking a red or blue marble are mutually exclusive

Probability of an Event

P(E)= The probability of an event.
n(E)= The number of outcomes of event E
n(S)= The number of outcomes in the sample space S.


     

where  




When P(E)=0, the event is an impossible event and cannot occur.
When P(E)=1, the even is a certain event and must occur.

Probability of the Union of Two Events

 -Means the intersection of two sets.
 -AKA "and"

-Means the union of two sets
-AKA "or"

When A and B are mutually exlusive (have no intersects):

     


When A and B aren't mutually exclusive (will work for all cases though):

Example: In this experiment you are rolling a die one time. What is the probability that the number rolled on the die is either even or prime?

The A circle of the venn diagram has all even numbers (2,4,6) and the B circle of the venn diagram has all prime numbers (2,3,5). 2 is common to both A and B. (1) is not a part of either circle because it is neither even nor prime.

                    

So the probability of rolling either an even or prime number is 5/6                                                                
                                 

Probability of Independent Events

If A and B are independent events, the probabilty that they will both occur is:




Example: You have ten socks in a drawer. 4 are blue, 5 are red, and 1 is green.  What is the probability that if you pick out 2 socks at random, what is the probability that the first sock will be blue and the second one green?

The probability of this happening is 2 in 45.

Probability of a Complement

If A is an event and is its complement then,

 

 

All done,
    Hannah