**Three types of Probability**

: For theoretical reasons, we assume that all**Theoretical probability**possible outcomes of a particular experiment are equally likely, and we assign a probability of*n*^{1}^{n}

to each possible outcome. Example: The theoretical probability of rolling a 3 on a regular 6 sided die is 1/6.

interpretation of probability: We conduct an experiment many, many times. Then we say**Relative frequency**

How many times A occurs

The probability of an event * A *=

Relative Frequency is based on observation or actual measurements.

Example: A die is rolled 100 times. The number 3 is rolled 12 times. The relative

frequency of rolling a 3 is 12/100.

: These are values (between 0 and 1 or 0 and 100%) assigned by individuals based on how likely they think events are to occur. Example: The probability of my being asked on a date for this weekend is 10%.**Personal or subjective probability**

**Probability Rules**

- The probability of an event is between 0 and 1. A probability of 1 is equivalent to

100% certainty. Probabilities can be expressed at fractions, decimals, or percents*.*

*0 ≤ pr(A) ≤ 1*

- The sum of the probabilities of all possible outcomes is 1 or 100%. If
,*A*, and*B*are the only possible outcomes, then*C**pr(A) + pr(B) + pr(C) = 1*

Example: A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles.

5 3 __2 __

*pr(red) + pr(blue) + pr(green) = 1 **+ + ** =* 1

10 10 10

- The sum of the probability of an event occurring and it not occurring is 1.
*pr(A) +*

*pr(not A) = 1 or pr(not A) = 1 – pr(A)*

.

Example: A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles.

__3 __

*pr (red) + pr(not red) = 1 **+** pr *(

*)*

*notred**1*

*=*

*pr (not red) =*__7__

10 10

- If two events
and*A*are independent (this means that the occurrence of*B*has no impact at all on whether*A*occurs and vice versa), then the probability of*B*and*A*occurring is the product of their individual probabilities.*B*

*pr (A and B) = pr(A) · pr(B)*

Example: roll a die and flip a __coin. __*pr**(heads and roll a 3) = pr(H) and pr(3)*

1 • 1 = 1

2 6 12

- If two events
and*A*are mutually exclusive (meaning*B*cannot occur at the same time as*A*occurs), then the probability of either*B*or*A*occurring is the sum of their individual probabilities.*B*)*Pr(A or B) = pr(A) + pr(B*

Example: A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles.

5 2 __7 __

**pr(red or green) = pr(red) + pr(green**)

*+ =*10 10 10

- If two events
and*A*are not mutually exclusive (meaning it is possible that*B*and*A*occur at the same time), then the probability of either*B*or*A*occurring is the sum of their individual probabilities minus the probability of both*B*and*A*occurring.*B**Pr(A or B) = pr(A) + pr(B) – pr(A and B)*

Example: There are 20 people in the room: 12 girls (5 with blond hair and 7 with brown hair) and 8 boys (4 with blond hair and 4 with brown hair). There are a total of 9 blonds and 11 with brown hair. One person from the group is chosen __randomly__*. pr**(girl or blond) = pr(girl) + pr(blond) – pr(girl and blond)*

12

+ 9 5 = 16

20 20 20 20

- The probability of at least one event occurring out of multiple events is equal to one minus the probability of none of the events
__occurring.__*pr*Example: roll a die 4 times. What is the probability of getting at least one head on the 4*(at least one) = 1 – pr(none)*

1 1 1 1

__rolls. __*pr***(at least one H) = 1 – pr(no H) = 1 – pr (TTTT) **= 1 –

*1-*

*• • • =*2 2 2 2

1 = 15

16 16

- If event
is a subset of event*B*, then the probability of B is less than or equal to the probability of A*A**. pr(B) ≤ pr(A)*

Example: There are 20 people in the room: 12 girls (5 with blond hair and 7 with brown hair) and 8 boys (4 with blond hair and 4 with brown hair). *pr (girl with*

12

*brown hair) ≤ pr(girl) *__7 __ £

20 20