HS 67BPS Chapter 101 Chapter 10 Introducing Probability.

HS 67 BPS Chapter 10 1

Chapter 10

Introducing Probability


Idea of Probability• Probability is the

science of chance behavior

• Chance behavior is unpredictable in the short run, but is predictable in the long run

• The probability of an event is its expected proportion in an infinite series of repetitions

The probability of any outcome of a random variable is an expected (not observed) proportion


How Probability BehavesCoin Toss Example

Eventually, the proportion of

heads approaches 0.5


How Probability Behaves“Random number table example”

The probability of a “0” in Table B is 1 in 10 (.10)

Q: What proportion of the first 50 digits in Table B is a “0”?

A: 3 of 50, or 0.06

Q: Shouldn’t it be 0.10?

A: No. The run is too short to determine probability. (Probability is the proportion in an infinite series.)


Probability models consist of two parts:1) Sample Space (S) = the set of all possible

outcomes of a random process. 2) Probabilities for each possible outcome in

sample space S are listed.

Probability Models

Probability Model “toss a fair coin”

S = {Head, Tail}

Pr(heads) = 0.5

Pr(tails) = 0.5


Rules of Probability


Rule 1 (Possible Probabilities)Let A ≡ event A

0 ≤ Pr(A) ≤ 1

Probabilities are always between 0 and 1.

Examples:

Pr(A) = 0 means A never occurs

Pr(A) = 1 means A always occurs

Pr(A) = .25 means A occurs 25% of the time


Rule 2 (Sample Space)Let S ≡ the entire Sample Space

Pr(S) = 1All probabilities in the sample space together must sum to 1

exactly.

Example: Probability Model “toss a fair coin”, shows that Pr(heads) + Pr(tails) = 0.5 + 0.5 = 1.0


Rule 3 (Complements)Let Ā ≡ the complement of event A

Pr(Ā) = 1 – Pr(A)

A complement of an event is its opposite

For example:

Let A ≡ survival then Ā ≡ death

If Pr(A) = 0.95, then

Pr(Ā) = 1 – 0.95 = 0.05


Events A and B are disjoint if they are mutually exclusive. When events are

disjoint

Pr(A or B) = Pr(A) + Pr(B)

Age of mother at first birth

(A) under 20: 25%

(B) 20-24: 33%

(C) 25+: 42% } Pr(B or C) = 33% + 42% = 75%

Rule 4 (Disjoint events)


Discrete Random Variables

Example: A couple wants three children. Let X ≡ the number of girls they will haveThis probability model is discrete:

Discrete random variables address outcomes that take on only discrete (integer) values


• Example Generate random number between 0 and 1 infinite possibilities.

• To assign probabilities for continuous random variables density models (recall Ch 3)

Continuous Random VariablesContinuous random variables form a continuum of possible outcomes.

This is the density model for random numbers between 0 and 1


Area Under Curve (AUC)The AUC concept (Chapter 3) is essential to working with continuous random variables.

Example: Select a number between 0 and 1 at random.

Let X ≡ the random value.

Pr(X < .5) = .5

Pr(X > 0.8) = .2


Normal Density Curves

→

z

x

♀ Height X~N(64.5, 2.5)

Z Scores

Introduced in Ch 3: X~N(µ, ).

Standardized Z~N(0, 1)

zx


If I select a woman at random a 99.7% chance she is between 57" and 72"

68-95-99.7 Rule• Let X ≡ ♀ height

(inches)• X ~ N (64.5, 2.5) • Use 68-95-99.7 rule

to determine heights for 99.7% of ♀

• μ ± 3σ = 64.5 ± 3(2.5)

= 64.5 ± 7.5 = 57 to 72


Calculating Normal Probabilities when 68-95-99.7 rule does not apply

Recall 4 step procedure (Ch 3)

A: State

B: Standardize

C: Sketch

D: Table A


Illustration: Normal Probabilities

What is the probability a woman is between 68” and 70” tall? Recall X ~ N (64.5, 2.5)

4.15.2

)5.6468(

z

z(70 64.5)2.5

2.2

A: State: We are looking for Pr(68 < X < 70)

B: Standardize

Thus, Pr(68 < X < 70) = Pr(1.4 < Z < 2.2)


Illustration (cont.)

C: Sketch

D: Table A: Pr(1.4 < Z < 2.2) = Pr(Z < 2.2) − Pr(Z < 1.4)= 0.9861 − 0.9192 = 0.0669

HS 67BPS Chapter 101 Chapter 10 Introducing Probability.

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