# STAT 1034 Lecture Notes - Lecture 4: Fair Coin, Random Variable, Sample Space

Day 16 (10/2):

Chapter 4: Probability

• 4.1 Randomness

o We call a phenomenon random if individual outcomes are uncertain but there is

nonetheless a regular distribution of outcomes in a large number of repetition.

▪ Ex: Toss a fair coin. (Only two possible outcomes). If in a trail we toss it

10,000 time, the proportion of tosses that gives a head approaches 50%.

• Two important notes:

o Need to toss the coin a large number of times to see that the

heads approaches 50%. A sort number of tosses may not

generate half heads.

o Each toss is independent of any other toss.

▪ 50% is the probability of a head or tail.

o Generalizing, the probability of any phenomenon in the proportion of times the

outcome would occur in a very long series of repetitions.

• 4.2 Probability Models

o A mathematical description of a random phenomenon is called a probability model.

▪ A probability model consists of

• A sample space, S

o The set of all possible outcomes

▪ Ex: Toss a fair coin 2 times and record the outcomes

in order (H=heads, T=tails)

• HH, HT, TH, TT

•

▪ Ex: Toss a fair coin 6 times and record the outcomes

in order (H=heads, T=tails)

•

▪ Ex: Suppose we only want to know the number of

heads n the 2 toses. We toss the coin twice and then

count the numder of heads

• HH = 2, HT = 1, TH = 1, TT = 0

o S = {0,1,2}

• An assignment of probability

o We need to assign probability to complete our model. To do

this we use events.

▪ An event is an outcome or set of outcomes. An event

is a subset of the sample space and events have

probability

• Ex: take the sample of tossing a fair coin

twice: S = {HH, HT, TH, TT}

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o The exactly one head is an event

Day 17 (10/4):

o Ex: We toss a fair coin twice and record the sequence of heads and tails. The

associated probability model is:

▪ Outcomes: {HH, HT, TH, TT}

▪ Probability: (25%, 25%, 25%, 25%)

• What is the probability of exactly one head?

o P(exactly on head) = P(HT or TH)

▪ P(HT) + P(TH)

▪ 25% + 25% = 50%

• What is the probability of at least one head?

o P(at least one head) = P(HH or HT or TH)

▪ P(HH) + P(HT) + P(TH)

▪ 25% + 25% + 25% = 75%

• Complement

o P(at least one head) + P(no heads)

▪ P(at least one head) = 1 – P(no heads)

• 1- P(TT)

• 1 – 25% = 75%

o Ex: Roll a fair die and observe the number of spots face up. What is the probability

that the face up number?

▪ A) 1

▪ B) odd

▪ C) 1,2,3,4 or 5

▪ D) 7

• Each face has 1/6 chance

o A) 1/6

o B) 3/6

o C) 1 – P(6) = 1 – 1/6 = 5/6

o D) 0/6

o Independence

▪ If A and B are Independent

• P(A and B) = P(A) x P(B)

o Multiplication rule for independent events.

• Ex. Toss a fair coin twice. What is the probability of 2 tails? Use the

multiplication rule.

o A = 1st toss is a tail

o B = 2nd toss is a tail

▪ P(A and B) = P(A) x P(B)

• (1/2)(1/2) = 25%

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• Ex. Toss UNFAIR coin twice and record the sequence of heads and

tails.

o It is unfair since heads appears three times as often as tails.

▪ Number of tails = a

▪ Number of heads = 3a

• P(H) = 3/4

• P(T) = 1/4

▪ P(H and H) = P(H)P(H)

▪ P(H and T) = P(H)P(T)

▪ P(T and H) = P(T)P(H)

▪ P(T and T) = P(T)P(T)

• Outcomes: HH, HT, TH, TT

• Probability: 9/16, 3/16, 3/16, 1/16

o What is the probability of at least one head?

▪ P(at least one head) = P(HH or HT or TH)

• P(HH) + P(HT) + P(TH)

• 9/16 + 3/16 + 3/16 = 15/16

o 4.3 Random Variables

▪ A random variable is a variable whose value is a numerical outcome of a random

phenomenon

• Ex: toss a fair coin twice and record the sequence of heads and tails, x =

number of heads

o X is the random variable with possible values of 0, 1, 2

▪ S = {HH, HT, TH, TT}

• Note: A random variable assigns a number/ value to

each outcome in the sample space

▪ When we move from probability to statistical influence we will concentrate on

random variables since in stats we are most often interested in numerical outcomes.

▪ Types of rv

• Discrete

o X has possible values that can be given in an ordered list (often

finite). The probability distribution of x lists the values another

probabilities

Value of x

...

P(x)

...

▪ 0 ≤ ≤ 1

▪

o Ex: What is the probability distribution of the random variable x that

counts the number of heads in the unfair coin toss

Value of x

1

2

P(x)

1/16

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## Document Summary

10,000 time, the proportion of tosses that gives a head approaches 50%: two important notes, need to toss the coin a large number of times to see that the heads approaches 50%. To do this we use events: an event is an outcome or set of outcomes. An event is a subset of the sample space and events have probability: ex: take the sample of tossing a fair coin twice: s = {hh, ht, th, tt, the exactly one head is an event. Day 17 (10/4): ex: we toss a fair coin twice and record the sequence of heads and tails. If a and b are independent: p(a and b) = p(a) x p(b, multiplication rule for independent events, ex. Use the multiplication rule: a = 1st toss is a tail, b = 2nd toss is a tail, p(a and b) = p(a) x p(b) (1/2)(1/2) = 25, ex. The probability distribution of x lists the values another probabilities.