More Probability Concepts

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MORE PROBABILITY CONCEPTS

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You may need to review these Basic Probability Concepts before doing this web exercise.

PROPERTIES OF PROBABILITIES
For any event E , 0≤P(E)≤1 .

For a finite sample space (i.e., an experiment with a finite number of outcomes):
a probability of 1 means the event will definitely occur;
a probability of 0 means the event will definitely not occur.

Notice that if the experiment is to randomly choose a real number from the interval [0,1],
then the probability of choosing (say) the number 0.5 is zero, and yet this outcome could happen.
This is why a finite sample space is needed in the above statement.

If S is the sample space, then P(S)= 1 .

For any events A and B , P(A∪ B)=P(A )+P(B) -P(A∩B ) .
If you were just to add the two probabilities, then the intersection gets added twice;
this extra probability must be subtracted.

Recall that E¯ denotes the complement of E .
For any event, either it happens, or it doesn't!
Therefore, P(E)+ P(E ¯)=1 .
Equivalently, P(E ¯)=1 -P(E) .

Events A and B are said to be mutually exclusive if A∩B= ∅ .
Recall that ∅ denotes the empty set.
Thus, two events are mutually exclusive when they have nothing in common; i.e., their intersection is empty.
Notice that for mutually exclusive events, P(A∪ B)=P(A )+P(B) .

MULTIPLICATION COUNTING PRINCIPLE
If there are F choices for how to perform a first act,
and for each of these F ways,
there are S choices for how to perform a second act,
then there are F⋅S ways to perform the acts in succession.
(The idea extends to more than 2 acts.)

The idea is illustrated by the diagram at right.
If there are 2 piles, with 3 in each pile,
then the total is 2⋅3= 6 .

EXAMPLE (pizza choices)
Suppose that a pizza shop offers 3 types of crust,
2 different types of cheese, and 7 different toppings.
A single-topping pizza consists of a choice of crust, a choice of cheese (if desired), and a choice of topping.
How many different single-topping pizzas are there?
Solution: There are 3 choices for the crust, 3 choices for cheese (none, type 1, type 2), and 7 choices for the single topping.
Thus, there are 3⋅3⋅ 7=63 possible single-topping pizzas.

On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.