PROBABILITY. Sample space S: The collection (the set) of all possible outcomes of an experiment.

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Event:

A collection of outcomes (a subset of the sample space) Example: a) A = {even number}

b) E1 = {2 red cards}, E2 = { one red and one

black}, E3 = {one K and one A}

c) B = {exactly one person said Y}

Simple event (Elementary outcome): It contains only one possible outcome.

a) {6} six is rolled b) {K♥,Q♠}

c) {YYY} = {all three said Y}

Operations on events:

If A, B are any two events then a new event may be formed as Symbol not A A A or B A ∪ B A and B A ∩ B Example:

a) not A = {not even number}= {odd number}

b) E₁ or E₂ = {two red} or {one red and one black}= {at least one red}

E₁ and E₃= { two red} and {one K and one A}

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c) Let C = { second person said Y} = {YYY,NYN,NYY,YYN}

B and C ={exactly one said Y} and {second person said Y} = {NYN}

B or C = {YNN,NYN,NNY,YYY, NYY,YYN}

Venn diagrams:

Events A and B are mutually exclusive (disjoint) if they have no out come in common.

Example: A = {two red}, C = {one ♣ and one♠}

Impossible event: no outcome is in it. (Symbol:O/ ) Example: A and C

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To any event E we may assign a number P(E), the probability of event E.

Its properties:

1. 0 ≤ P(E) ≤ 1

2. If P(E) = 1, then it is a sure event.

3. If P(E) = 0, then it is an impossible event

Assignment of a probability to an event: a) classical

b) subjective

c) relative frequency

a) If all outcomes are equally likely, and there are in total N outcomes, then we assign probability

N

1

to each simple outcome. For example, there are 6 simple outcomes if you roll a die, and if all are equally likely (it is a ‘fair’ die), then the probability of rolling a 6 is

6 1

.

b) It is based on subjective judgement. For example one can assign a number 0 ≤ P(it will snow tonight) ≤ 1, based on experience.

c) If an experiment has been observed many times, the relative frequency of an event can be used as its probability. For example, to verify that a die is ‘fair’, one may roll it 1000 times, and if the relative frequency is a number

1000 200

, then we will assign this number as a probability of getting a 6.

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RULES:

I. P(A) is the sum of probabilities of all simple events that comprise A

II. P(not A) = 1 – P(A)

III. P(A or B) = P(A) + P(B) – P(A and B)

IV. If A and B are disjoint, then P(A or B) = P(A)+P(B) Examples:

a) Choose a card from a deck of well shuffled deck of cards. P( getting a ♥) = 52 13 = 4 1 P( K) = 52 4

b) Toss a fair coin once: P(H) = ½, P(T) = ½

Toss it twice: P(HH) = ¼, P(HT) = ¼, P(TH) = ¼, P(TT) = ¼

P(A) = P(at least one H) = {(HH),(HT),(TH)} = ¾ P(B) = P(no H) = P( not A) = 1-P(A) = 1-3/4 = ¼ Note, that P(A) + P(not A) = 1

c) What is the probability that a card picked at random will be a ♥ or a Q?

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A = {card is ♥}, B = { card is Q} P(A) = 13/52 , P(B) = 4/52

P (A or B) = P(A) + P(B) – P(A and B) P( A and B) = P( the cards is ♥Q} = 1/52 So P(A or B) = 13 /52 + 4 / 52 –1 / 52 = 13 4 52 16 52 1 4 13 = = − +

d) Draw two cards with replacement, event A = {both are red}, event B = {one is red the other black}

P(A or B) = P(A) + P(B) = 4 3 2 1 4 1 ₊ ₌

To verify: P(A or B) = P(not both black) =

4 3 4 1 1− =

CONDITIONAL PROBABILITY AND INDEPENDENT EVENTS

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The Conditional Probability of an event A, given the occurrence of an event B is defined by

P(A⎜B) =

P(_PAandB₍_B₎ )

, P(B) ≠ 0

when P(B) = 0, the conditional probability is not defined.

Example:

Roll a die. What is the probability of rolling 2 (event A), when you know you rolled an even number (event B)?

P(A) = 1/6, P(B) = 3/6, P(A and B) = 1/6

P(A⎜B)=1₃/_/6₆ = 1₃

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If A and B are two events, then

P(A and B) = P(A|B) P(B), if P(B) ≠ 0

Example:

1. The probability that an adult male is over 6 feet tall is 0.3 (say). Given that he is over 6 feet tall, the probability that he weighs less than 180 pounds is 0.2. Find the probability that a randomly chosen man will be taller than 6 feet and weighs less than 180 pounds.

Event A: a male is over 6 ft

Event B: he weighs less than 180 lb.

P(A and B) =

Two events a and B are independent if and only if

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Or, equivalently

P(A⎜B) = P(A), P(B) ≠ 0 Example:

Take 2 cards with replacement. Probability of 2 K ?

Event A: first is K, event B the second id K. P(A and B)=? Similarly, events A1, A2, …, Ak are independent if and only

if

P(A1 and A2 and …and Ak ) = P(A1) P(A2) …P(Ak)

Example:

Take 3 cards with replacement. Probability of 3K? Event A1: the first is K

event A2: the first is K

event A3: the first is K

P(A1 and A2 and A3) = P(A1) P(A2) P(A3) = ₅₂

4 52 4 52 4 Examples:

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given detection, a 50% chance of scaring the intruder away. What is the probability that Fido successfully thwarts a burglar?

2. A husband is late to work with probability 0.1, his wife is late to work with probability 0.2, and they are both late with probability 0.015. Can we say they are late independently?