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

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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
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.

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