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.
