3.4 Independent Events

 

Defn: A compound event is a situation involving two or more individual events.

 

Defn: Two events are said to be independent if the occurrence of one has no effect on the occurrence of the other.

 

Examples:

-         Rolling a six sided die and flipping a coin.

-         Repeated trials of flipping a coin

-         Spins on a Roulette Table

 

Ex 1:

 a) A coin is flipped and turns up heads. What is the probability that the second flip will also turn up heads?

b) A coin is flipped four times and lands heads each time. What is the probability that the fifth flip will land heads?

 

Product Rule for Independent Events

                        P(A and B) = P(A) *P(B)

 

Ex 2

A coin is flipped while a die is rolled. What is the probability of the coin landing heads and the die rolling a 5?

 

Proof of the Product Rule for Independent Variables

 

Since A and B are separate events they have separate sample spaces, SA and SB. The sample space for our compound events is then S = (SA and SB).

 

Consider example 2 above. What are the sample spaces? What are their sizes?

 

P(A and B) = n(A and B) = n(A and B)

 

Using the fundamental Counting Principle n(A and B) = n(A) * n(B) on both numerator and denominator gives:

 

 

 

 

Dependent Events

 

Defn: Two events are said to be dependent if the occurrence of one effect the occurrence of the other.

 

Examples:      - Doing well on a test and completing your homework?

                     - Drawing marbles from a bag (without replacement)

 

Defn: The conditional probability, denoted P(A|B) is the probability of the event A given that the event B has already occurred.

                       

If we know nothing about the event B what is the sample space? If we then find out that B has in fact occurred, how does this change our sample space?

 

Now our reduced sample space SA|B becomes B, thus our new Venn diagram is

And so

                                    n(A|B)/n(S)      = n(A and B)/n(B)

Dividing top and bottom by n(S) gives

                                                           = P(A and B) / P(B)

 

Product Rule for Dependent Events

                                    P(A and B) = P(A|B) * P(B)

 

Ex 3 What is the probability of rolling a sum greater than 7 given that one of the die rolls is a 3?

 

Ex 4 George is undecided as to whether to take a French or Chemistry course. He estimates that the probabilities of getting an A in each of the respective courses are ½ and 1/3. If he chooses which course to take based on the flip of a coin, what is the probability that George will get an A in Chemistry?

 

Homework: pg 334 #1-7, 9-11, 14