Probability

Likelihood of an event occurring!

Random CircumstancesA random circumstance

is one in which the outcome is unpredictable.

Test results are positive, but does that guarantee the person has the disease.

Someone’s name in class will be drawn to answer a question.

Flipping a CoinWhat are the outcomes

when flipping a fair coin?

What is the likelihood of each event?

If we flip the coin again, does the result of the first flip influence the result of the 2nd flip?

Rolling a dieWhat are the possible

outcomes when rolling a fair die?

What is the likelihood of each of the outcomes?

What is the sum of all the events’ probabilities?

If we rolled the die 600 times, what type of distribution would we see? Sketch graph.

Vocabulary

EventSample space

Relative frequencyLaw of Large Numbers

Conditional probability vs. Independent Events

Sample Space

Rolling a die…{ 1, 2, 3, 4, 5, 6 }Flipping two coins…{HH, HT, TH, TT}

List all possible outcomes from event

Conditions for Valid Probabilities

Each probability of an event has value from 0 to 1.

P(tails) = .5

P(roll a 4) = 1/6

P(today is Friday) = _?_

Sum of all probabilities of events in a sample space is 1.

P(heads) + P(tails) = 1

More VocabularyComplementary Events:

compliment of event A is AC, because

P(A) + P(AC) = 1

Mutually Exclusive Events: 2 events are disjoint because they do not contain any of the same events and can not occur simultaneously.

Complimentary event of obtaining tails when flipping a coin is the event of obtaining heads.

The set of events where all lottery digits are the same is mutually exclusive of the set of events of the first and last lottery digits being different.

More VocabularyRolling a fair dice is an example

of independent events. Knowledge of what the first roll was has no influence on the next roll.

If I were to randomly select students in class of 100 students without replacement, the probability of you being selected changes with each name called. 1/100, 1/99, 1/98,…0.

Independent Events: knowing one event has occurred does not change the probability of the other event occurring.

Dependent Events: knowing one event has occurred changes the probability of the other event occurring. Conditional Probability

Independent Events

P( 6 and 6) = P(6)P(6)

= (1/6)(1/6)

= 1/36

Dependent Events

P( 7and 6 w/no replacement) == P(7 )P(6 given 7 was drawn)= (1/52)(1/51)

Conditional Probability

Find the probability of drawing a red ball out of a box containing 3 red, 4 blue, and 1 white GIVEN that a blue ball has been drawn and not replaced.

P(R|B) = 3/7

The probability of having a certain disease is .05. The probability of testing + if you have the disease is .98; the probability of testing + when you do not have it is .10. What is the probability that you have the disease if you test +?

3403.

10.98.05.

98.05.

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Addition Rule

Find the probability that you draw either an ace or a red card.

P(Ace or Red) = P(Ace) + P(Red) – P(Red Ace)

= (4/52) + (26/52) – (2/52)

= 28/52

Mutually exclusive (Disjoint): P(A & B) = 0

Rules0 P(event) 1

Complement P(eventC) = 1 – P(event)P(S) = 1, where S is Sample Space

If A & B are independent, then P(A&B)=P(A)P(B)

Conditional: If A & B are dependent, then P(A&B)=P(A)P(B given A occurred)

P(A&B) = P(A)P(B|A)

Sample Exam Questions

Tablets Female Male Total

1 2 3 5

2 4 6 10

3 2 2 4

4 0 1 1

Total 8 12 20

Who is more likely, male or female, to take 2 tablets?

What is the probability that a person took only 1 tablet?

Sample Exam Questions

Which of the following is NOT something we would know from a probability?

a) How likely a specific outcome is to occur

b) When a specific outcome will occurc) How often a specific outcome will occur

in the long rund) We don’t know any of the abovee) We would know a, b, and c.

Sample Exam Questions

How can we determine probabilities?

a) Through simulation.

b) From a random sample.

c) From a mathematical formula/equation.

d) All of the above.

e) Exactly 2 of the above are true.