20 3.3 Two Basic Rules of Probability
When calculating probability, there are two rules to consider when determining if two events are independent or dependent and if they are mutually exclusive or not.
The Multiplication Rule
If A and B are two events defined on a sample space, then: P(A AND B) = P(B)*P(A|B).
This rule may also be written as [latex]\displaystyle{P}{({A}{\mid}{B})}=\frac{{{P}{({A}\text{ AND } {B})}}}{{{P}{({B})}}}[/latex]
(The probability of A given B equals the probability of A and B divided by the probability of B.)
If A and B are independent, then P(A|B) = P(A).
Then P(A AND B) = P(A|B)*P(B) becomes P(A AND B) = P(A)*P(B).
The Addition Rule
If A and B are defined on a sample space, then: P(A OR B) = P(A) + P(B) – P(A AND B).
If A and B are mutually exclusive, then P(A AND B) = 0. Then P(A OR B) = P(A) + P(B) – P(A AND B) becomes P(A OR B) = P(A) + P(B).
Example 1
Klaus is trying to choose where to go on vacation. His two choices are: A = New Zealand and B = Alaska.
Klaus can only afford one vacation.
The probability that he chooses New Zealand is 0.6 and the probability that he chooses Alaska is 0.35.
Klaus can only afford to take one vacation.
- What is the probability that he chooses either New Zealand or Alaska?
- What is the probability that he does not choose to go anywhere on vacation?
Show Answer
- Let A be New Zealand, B be Alaska.
P(Klaus chooses New Zealand) = P(A) = 0.6
P(Klaus chooses Alaska ) = P(B) = 0.35
P(Klaus chooses both New Zealand and Alaska) = P(A and B) = 0 as he can only afford one vacation.
P(A OR B) = P(A) + P(B) – P(A and B) = 0.6 + 0.35 – 0 = 0.95.
Therefore, the probability that he chooses either New Zealand or Alaska is 0.95. - The probability that he does not choose to go anywhere on vacation
= 1 – P( A or B)
= 1 – 0.95
= 0.05.
Example 2
Carlos plays college soccer. He makes a goal 65% of the time he shoots. Carlos is going to attempt two goals in a row in the next game.
A = the event Carlos is successful on his first attempt.
B = the event Carlos is successful on his second attempt.
Carlos tends to shoot in streaks. The probability that he makes the second goal GIVEN that he made the first goal is 0.90.
P(A) = 0.65, P(B) = 0.65, P(B|A) = 0.90.
- What is the probability that he makes both goals?
Show Answer
The problem is asking you to find P(A AND B) = P(B AND A).
Since P(B|A) = 0.90, P(B AND A) = P(B|A) * P(A) = (0.90)(0.65) = 0.585
Carlos makes the first and second goals with probability 0.585. - What is the probability that Carlos makes either the first goal or the second goal?
Show Answer
The problem is asking you to find P(A OR B).
P(A OR B) = P(A) + P(B) – P(A AND B) = 0.65 + 0.65 – 0.585 = 0.715
Carlos makes either the first goal or the second goal with probability 0.715. - Are A and B independent?
Show Answer
P(B AND A) = 0.585. P(B)P(A) = (0.65)(0.65) = 0.423
Since P(B AND A) [latex]\ne[/latex] P(B)*P(A), A and B are not independent. - Are A and B mutually exclusive?
Show Answer
P(A and