Definitions: Probability

In *Probability*, we are talking about the likelihood that some *event* will occur.

The probably of some event is written as ... and runs from 0% (the event will *never* happen) to 100% (the event will *always* happen). It may be written as decimals, from 0 to 1, or as fractions. A probability will *never* be negative or more than 1.

The set of ways something can happen is the *sample space*.

The event we're interested in are the *outcome*.

Definitions: Probability

If we are flipping a coin and we've called Heads, the sample space is {Heads,Tails} and the outcome is {Heads}.

If we are rolling a six-sided die and we're looking to get a number more than 2, the sample space is {1,2,3,4,5,6} and the outcome is {3,4,5,6}.

Calculating Simple Probability

If all outcomes are equally likely, you can calculate the probability of an event occurring using:

...

Calculating Simple Probability

- What is the probability of flipping a coin and getting heads?
As above, the sample space is {Heads,Tails}, which has 2 elements, and the outcome is {Heads}, which has 1.

So the probability of getting heads is ...

- What is the probability of rolling a 6-sided die and getting a number more than 4?
The sample space for rolling a 6-sided die is {1, 2, 3, 4, 5, 6}, which has 6 elements.

The outcomes we're interested in, the numbers more than 4, are {5, 6}, which has 2 elements.

So the probability of rolling a number more than 4 is ...

- What is the probability of flipping a coin
*twice*and getting heads*both times*? - What are the possible ways we can flip a coin twice? {HH, HT, TH, TT} has 4 elements. There is only one way to get heads both time: {HH} has 1 element.
So the probability of getting heads twice is ...

- Define
*Probability*as the likelihood of an*event*occurring. It ranges from 0% (will never happen) to 100% (will always happen). It's often referred to using decimals, from 0-1, or as fractions. - Define a
*random experiment*as one in which the*outcomes*are based on chance. For example, flipping a coin has two possible outcomes: Heads or Tails; that set of outcomes is the*sample space*. If we're only interested in getting Tails, then that is the*event*we're talking about.If we're rolling a six-sided die, and we're only interested in getting even numbers, then the sample space is {1,2,3,4,5,6}, the event is "rolling an even number," and the outcomes are {2,4,6}.

- If all outcomes are equally likely, we can find the probability of some event using .
- Tree diagrams for more complicated sample spaces.
- If all outcomes are
*not*equally likely, you can find the probability by counting. - Venn diagranms.
- Define the
*complement*of an event as its opposite. The complement of A is A^{c}; if A is "has a sister" then A^{c}is "does not have a sister."These form the entire sample space: P(A) + P(A

^{c}) = 1, or P(A^{c}) = 1 - P(A). - An
*intersection*of events represents an intersection: Who has a sister*and*a cat? Written P(A [intersect] B). See overlap in Venn Diagrams. - A
*union*represents a combination: Who has a sister*or*a cat? Written P(A U B) [for union]. See Venn Diagram - [Screen 17] Geometric probability uses area: Area of event / Area of sample space.
- Define a
*fair decision*as one in which the outcomes are chosen at random, and each has the same probability.