for student 2

Probability

Introduction to Probability,

Conditional Probability

Why do we need Probability?

• We have several graphical and numerical statistics for summarizing our data

• We want to make probability statements about the significance of our statistics

• Eg. In Stat111, mean(height) = 66.7 inches

• What is the chance that the true height of the university’s students is between 60 and 70 inches?

• Eg. r = -0.22 for draft order and birthday

Deterministic vs. Random Processes

• In deterministic processes, the outcome can be predicted exactly in advance

• Eg. Force = mass x acceleration. If we are given values for mass and acceleration, we exactly know the value of force

• In random processes, the outcome is not

known exactly, but we can still describe the

probability distribution

Sample Space

• The sample space S of a random process is

the set of all possible outcomes Example: one coin toss

S = {H,T}

Example: three coin tosses

S = {HHH, HTH, HHT, TTT, HTT, THT, TTH, THH}

Example: roll a six-sided dice

S = {1, 2, 3, 4, 5, 6}

Example: Pick a real number X between 1 and 20

Events

• An event is an outcome or a set of outcomes of a random process

Example: Tossing a coin three times

Event A = getting exactly two heads = {HTH, HHT, THH}

Example: Picking real number X between 1 and 20

Event A = chosen number is at most 8.23 = {X ≤ 8.23}

Example: Tossing a fair dice

Combinations of Events

• The complement Ac _{of an event A is the event that A}

does not occur

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

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

Equally Likely Outcomes Rule

• If all possible outcomes from a random process have the same probability, then

• P(A) = (# of outcomes in A)/(# of outcomes in S)

• Examples: One Dice Tossed, find P(A) if: A: the outcome is even number

A: the outcome does not equal 6

• Examples: Two coins tossed, find P(A) if: A: the outcome contains at least one head

Probability Rules

•

P(S) = 1

•

0 ≤ P(A) ≤ 1 for any event A

•

P(A

) = 1 - P(A)

• Examples: Two coins tossed, find P(A) if: A: the outcome contains at least one head

Disjoint Events

• Two events are called disjoint if they can not

happen at the same time

• Events A and B are disjoint means that the intersection of A and B is zero

• Example: coin is tossed twice

• S = {HH,TH,HT,TT}

• Events A={HH} and B={TT} are disjoint

• Events A={HH,HT} and B = {HH} are not disjoint

•

Rule 4:

Independent events

• Events A and B are independent if knowing that A occurs does not affect the probability that B occurs

• Example: tossing two coins

Event A = first coin is a head Event B = second coin is a head

• Disjoint events cannot be independent!

• If A and B can not occur together (disjoint), then knowing that A occurs does change probability that B occurs

• Probability Rule 5: If A and B are independent P(A and B) = P(A) x P(B)

Independent

• Example 1:

A- Tossing one dice and one coin, find the following: • 1- The sample space

• 2-The event A: the dice is even number and the coin is a head

• 3- P(A)

B- If A and B are two events where P(A)=0.3, P(B)=0.4, and P(A∩B)=0.12,

• Example 2 Suppose you roll a red and a green dice. Find

1. P(sum is 4)

2. P(sum is at most 4)

6. P(the red dice shows a 6)

7. P(both dice show 6)

8. P(sum of dice exceeds 1)

9. P(sum of dice exceeds 12)

Definition: Conditional Probability

• Let A and B be two events in sample space. The

conditional probability that event B occurs

given

that event A has occurred

Independent vs. Non-independent Events

• If A and B are independent, then

P(

A and B

A

B

which means that conditional probability is:

P(B | A) = P(A and B) / P(A) = P(A)P(B)/P(A) = P(B)

• We have a more general multiplication rule for

events that are not independent:

Random variables

• A random variable is a numerical outcome of

a random process or random event

• Example: three tosses of a coin

• S = {HHH,THH,HTH,HHT,HTT,THT,TTH,TTT}

• Random variable X = number of observed tails

• Possible values for X = {0,1, 2, 3}

• Why do we need random variables?

Discrete Random Variables

• A discrete random variable has a finite or

countable number of distinct values

• Discrete random variables can be summarized by listing all values along with the probabilities

• Called a probability distribution

• Example: number of members in US families

x 2 3 4 5 6 7

Another Example

• Random variable X = the sum of two dice

• X takes on values from 2 to 12

• Use “equally-likely outcomes” rule to calculate the probability distribution:

• If discrete r.v. takes on many values, it is

better to use a probability histogram

x 2 3 4 5 6 7 8 9 10 11 12

# of

Outcomes 1 2 3 4 5 6 5 4 3 2 1

Probability Histograms

• Probability histogram of sum of two dice:

• Using the disjoint addition rule, probabilities for discrete random variables are calculated by adding up the “bars” of this histogram:

Another Example

• Example:

Random variable X = the number of heads in tossing two coins, find the probability distribution and

histogram

x

Example: find the expected sum for rolling two dice

X = sum of two dice

x 2 3 4 5 6 7 8 9 10 11 12 P(X=x

Example: find the variance for the sum of two dice

X = sum of two dice

x 2 3 4 5 6 7 8 9 10 11 12 P(X=x