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Section #6

November 03, 2008

TA: Victoria Liublinska

vliublin@fas.harvard.edu

Homework & Midterm commentsMidterm comments:

Solution is posted online Distribution of scores:

mean 63.4SD 7.3median 65

Initial Project Proposals!

Outline for today:

What is probability? Terminology and rules Disjoint vs. independent events Example of calculating probabilities and

expectation for a random variable

Interpretations of Probability

Probability: Long run frequency

We can think of it as a long run average of the number of times an event occurs in e sequence of independent trials

It is applicable only when the situation might be repeated numerous times (at least conceptually)

The outcomes have same chances every time (no time dependence)

We still can’t predict individual outcome, only the long-run average

Personal Probability (Decision making)

Practice Problem Suppose a particular outcome from a random event has a probability of 0.02. Which of the following statements represent correct interpretations of this probability? Provide justification.

i) The outcome will never happen.

ii) The outcome will happen two times out of every 100 trials, for certain.

iii) The outcome will happen two times out of every 100 trials, on average.

iv) The outcome could happen, or it couldn't, the chances of either results are the same.

Probability Terminology

Terminology random phenomenon – an event whose individual

outcomes are uncertain but there is a regular distribution in a large number of repetitions. Examples:

Coin tossing and dice rolling The lottery and other games of chance Drawing a random sample from some population

outcome: the value of one replication of a random experiment or phenomenon, Coin Tossing:

H with one toss of a coin HTT with three tosses

Probability Terminology (cont.)

sample space (labeled S): is the set of all possible outcomes of a random phenomenon

Examples: 1. Toss a coin three times: S = {HHH,THH,HTH,…,TTT}

2. Face showing when rolling a six-sided die: S = {1,2,3,4,5,6}

3. Pick a real number between 1 and 20: S ={[1,20]} Event (labeled A): a set of outcomes of a random phenomenon. Examples:

1. The event A that exactly two heads are obtained when a coin is tossed three times: A={HHT,HTH,THH}

2. The number chosen from the set of all real numbers between 1 and 20 is at most 8.23: A = {[1,8.23]}

3. The result of the toss of a fair die is an even number:A = {2,4,6}

Events in Sample Spaces(more Terminology)

The union of two events A and B is the event that either A occurs or B occurs or both occur:

The intersection of two events A and B is the event that both A and B occur.

( or )C A B A B

The complement of an event A, Ac, is the event that A does not occur and thus consists of outcomes that are not in A

Probability rules

Example – mandatory drug testsA mandatory drug test has a

false-positive rate of 1.2% (or 0.012)

Given 150 employees who are drug free, what is the probability that at least one will (falsely) test

positive?

P(At least 1 positive) = P(1 or 2 or 3 ... or 150 “+”) = 1 – P(None “+”) = 1 – P(150 “-”)

P(150 negative) = P(1 “-”)150 = (0.988)150 = 0.16

P(At least 1 positive) = 1 – P(150 “-”) = 0.84

Disjoint vs Independent Two events are independent if the occurrence of one of the

events gives us no information about whether or not the other event will occur; that is, the events have no influence on each other (no INTERACTION between events). (me/you being in Boston)

Two events are disjoint (or mutually exclusive) if it is impossible for them to occur together (not COMPATIBLE) (me being in Boston or Las Vegas)

Connection: If two events are mutually exclusive, they cannot be independent and vice versa (if two events are independent then they cannot be mutually exclusive (disjoint)). (If I am in Boston – I know that I am not in Las Vegas)

Example

There is a bag with 4 balls in it: 2 are red, and 2 are black

You draw two balls out of the bag, one at a time. Define the events:

A: the first ball drawn is black

B: the second ball drawn is black

What is the probability of A ? Are A and B independent?

Independent events

Rule 5: Two events A and B are independent if knowing that one occurs does not change the probability that the other occurs. If A and B are independent P(A and B) = P(A) P(B) Sometimes called the multiplication rule for

independent events. Does knowing the results of flipping a fair coin once affect

the chances of heads on a 2nd flip?

Mean of a random variable

If X is a discrete random variable with k possible values, its probability distribution is:

Value x1 x2 x3 ... xk

Probability p1 p2 p3 ... Pk

Mean (or “expected value”) of X, denoted μX, is given by

E(X) = X = x1 p1 + x2 p2 + ... + xk pk

If X represents a measure on some member of a population, the E(X) is the population mean

Example

A lottery has 100 tickets. Each ticket costs $10. Only one ticket is the “lucky” one and the cash prize for it is $500. Is it worthwhile for Mary to buy a lottery ticket?

Answer the question by computing Mary’s expected gain

Practicing with eventsI have 5 M&M’s left in the bag: 2 blue, 2 red, and 1 yellow. I plan on eating two more, and leaving the others in the bag.

1. What is the sample space of outcomes for pairs of M&M’s that I will eat?

2. Describe the set of events for: A: eating two mismatched M&M’s (different colors).B: eating at least one blue M&M (they are my favorite).3. What is the probability of A? B?4. Are A & B disjoint? Why?5. Describe (in terms of outcomes), D, the intersection of A & B? and

the union, E?6. What is the probability of D? of E? 7. Are A & B Independent? Why?

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Excel Demonstration