Probability_Fundamentals

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Three types of Probability

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

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

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

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.

  1. Personal or subjective probability: 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%.

Probability Rules

  1. 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

  1. The sum of the probabilities of all possible outcomes is 1 or 100%. If A, B, and C are the only possible outcomes, then 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

  1. 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

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

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

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

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

  1. 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(at least one) = 1 – pr(none) Example: roll a die 4 times. What is the probability of getting at least one head on the 4

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

  1. If event B is a subset of event A, then the probability of B is less than or equal to the probability of 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

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