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### Lesson 6

Probability

Probability Everyday

Probability is an everyday occurrence in our lives.

- What is the probability it will rain today?
- What is the probability you will get a 90% or better on your mid-term?
- What is the probability you will win the super lottery?

Use Common Sense

- When approaching a probability problem, it is usually best to use your common sense. What do you expect would happen in a given situation?
- For example, if I give you a die and tell you to throw it, what is the probability that you will roll a 5?

Probability Formula

The general formula for finding the probability of an event is:

Number of outcomes in the event

Total number of possible outcomes

p(A) =

Probability Facts

The probability of an event is always between 0 and 1.00. That is,

- If an event can never happen then p(A) = 0.
- If an event always happens then p(A) = 1.00

Useful Terms

The following are some useful terms when doing probability problems:

- Random sample
- Sampling with replacement
- Sampling without replacement
- Mutually exclusive
- Independent events
- Dependent events

Random Sample

- In a random sample:
- Each individual in the population has an equal chanceof being selected.
- If more than one individual is to be selected for the sample, there must be constant probabilityfor each and every selection.

Sampling

- In sampling with replacement, an individual selected is returned to the population before the next selection is made.
- In sampling without replacement, an individual selected is not returned to the population before the next selection.

Mutually Exclusive

- Two events are mutually exclusive if they cannot occur simultaneously. For example:
- A single roll of a die cannot result in a 2 AND a 5.
- A single card selected from a deck cannot be a Heart AND a Diamond (mutually exclusive), but it can be a Heart AND a Queen (not mutually exclusive).

Independent Events

Two events are independentif the outcome of one event does not effect the probability of the second. For example:

- Rolling a single die twice (or rolling two dice simultaneously) are independent events. What you get on one roll does not effect the second roll.
- Drawing two cards from a deck with replacement.

Independent Events

- Conclusion—drawing two cards out of a deck without replacement are NOT independent events.

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Dependent Events

Two events are dependentif the outcome of one event does effectthe probability of the second. For example:

- Drawing two cards from a deck without replacement.

Addition Rules

- General Addition Rule for finding p(A or B):
p(A or B) = p(A) + p(B) – p(A and B)

- When A and B are mutually exclusive
p(A or B) = p(A) + p(B)

Addition Rules

- General Addition Rule for finding p(A or B):
p(A or B) = p(A) + p(B) – p(A and B)

- p(diamond) = 13/52
- p(5) = 4/52
- p(5 of diamonds) = 1/52
- p(diamond OR 5) = 13/52 + 4/52 – 1/52 = 16/52

Multiplication Rules

General Multiplication Rule for finding p(A and B):

p(A and B) = p(A)p(B|A) where p(B|A) is the probability of event B given that event A has already occurred.

When A and B are independent events:

p(A and B) = p(A)p(B)

When A and B are mutually exclusive, p(A and B) = 0

Multiplication Rules

p(A and B) = p(A)p(B|A) where p(B|A) is the probability of event B given that event A has already occurred.

- Event A is drawing a King
- Event B is drawing a 5
- p(A) = 4/52
- p(B|A) = ¼
- p(A and B) =

/

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Continuous Probability

The formula to find the probability of an event from a continuous normally distributed variableis:

X - m

s

z =

Probability and Proportion

- The good news is that the probability of an event, that is a single score or a set of scores, from a normal distribution is exactly the same thing as the proportion. Use exactly the same procedures, formulas, etc. that you used before. Just refer to your answer as the probability of the event.

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