# Unit 12. Applications of Probability

## 12.1: What Is Probability?

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 Ac; if A is "has a sister" then Ac is "does not have a sister."

These form the entire sample space: P(A) + P(Ac) = 1, or P(Ac) = 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.