Section 4.1
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Coin Flips
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Roadmap
Section 4.1 establishes terms for probability distributions. You will see
familiar tables and mathematics, as we will see the first terms are similar
to frequency distributions.
I
Random variables: discrete and continuous.
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Rules for discrete probability distributions.
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Mean and variance (and standard deviation) of discrete probability
distributions.
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Random Variables
A random variable is a process that turns an outcome of a probability
experiment into a numerical value.
Example The sample space for a coin toss is S = {H, T }. A random
variable could be x = 1 if the toss is heads, and x = 0 if the toss is tails.
Outcome Event E
T
H
Random Variable Value x
0
1
Probability P(x)
0.5
0.5
In this example, x counts once if there is a head, and no times if there is a
tail.
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Random Variables
Example Two coins are tossed. The random variable x counts the number
of heads.
The sample space is S = {HH, HT , TH, TT }.
Outcome Event E
TT
HT or TH
HH
Random Variable Value x
0
1
2
Probability P(x)
0.25
0.50
0.25
The probability distribution for the random variable is then
Number of heads
0
1
2
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Probability
0.25
0.50
0.25
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Random variable
There are two types of random variables: discrete and continuous. In most
settings, discrete r.v.’s represent counted data, and continuous r.v.’s
represent measured data.
Discrete random variables are when a finite or countable number of
outcomes can be listed.
Continuous random variables are when there is an uncountable number
of outcomes. This is represented by an interval or number line.
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Random Variables
Examples of discrete random variables include:
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Number of children delivered in a hospital per day.
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Number of successful attempts at a matching quiz.
Examples of continuous random variables include:
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The volume of oxygen in a balloon.
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The amount of minutes spent on a phone call.
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Discrete Probability Distribution
Discrete probability distributions list the values of a random variable, along
with its probability. It must:
1. Have the probability of each value between 0 and 1, inclusive.
2. The sum of all the probabilities must be 1.
In symbols,
1. 0 ≤ P(x) ≤ 1.
P
2.
P(x) = 1.
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Discrete Probability Distribution
Probabilities are relative frequencies. A discrete probability distribution is
the relative frequency distribution, and they can be represented by relative
frequency histograms.
Example Let x be the sum of two six sided dice.
Frequency distribution:
x
f
2
1
3
2
4
3
5
4
6
5
7
6
8
5
9
4
10
3
11
2
12
1
Probability distribution:
x
P(x)
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2
3
4
5
6
7
8
9
10
11
12
1
36
2
36
3
36
4
36
5
36
6
36
5
36
4
36
3
36
2
36
1
36
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Discrete Probability Distribution
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Mean / Expected Value
The mean of a discrete random variable is:
µ=
X
x · P(x)
This is the same as the expected value E (x).
Example For two coin flips, the average number of heads is:
1
1
1
µ=0·
+1·
+2·
=1
4
2
4
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Mean / Expected Value
Example A raffle has 900 tickets, sold at $4 each. There are four prizes:
$500, $400, $200, and $100. You buy 1 ticket. What is the expected value
of your gain?
Gain is the prize value minus the ticket value.
Top prize gain: $500 − $4 = $496,
Losing ticket: $0 − $4 = −$4
Probabiliy distribution:
Gain, x
P(x)
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$496
$396
$196
$96
-$4
1
900
1
900
1
900
1
900
896
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Mean / Expected Value
Gain, x
P(x)
E (x) =
X
$496
$396
$196
$96
-$4
1
900
1
900
1
900
1
900
896
900
x · P(x)
1
1
1
896
1
+ $396 ·
+ $196 ·
+ $96 ·
+ (−$4) ·
900
900
900
900
900
= −$2.67
= $496 ·
The expected value is negative, so you can lose an average of $2.67 for
each ticket you buy.
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Variance and Standard Deviation
Outcomes vary, so it is useful to seek a probability distribution’s variance
and standard deviation.
The variance: σ 2 =
X
(x − µ)2 · P(x).
√
The standard deviation: σ =
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σ2 =
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qX
(x − µ)2 · P(x).
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Variance and Standard Deviation
Example For two coin flips,
x
0
1
2
P(x)
0.25
0.50
0.25
P
P(x) = 1
x −µ
-1
0
1
(x − µ)2
1
0
1
(x − µ)2 · P(x)
0.25
0
0.25
P
(x − µ)2 · P(x) = 0.50
The √
variance is 0.50 number of heads, squared. And standard deviation is
σ = 0.50 ≈ 0.7071 number of heads. Most of the data values will differ
from the mean by no more than 0.7071 heads.
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Assignments
Assignment:
1. Read pages 190 - 196.
2. Exercises 1 - 45 odd on page 197.
Vocabulary: random variable, continuous and discrete random variable,
probability distribution, mean, expected value, variance and standard
deviation
Understand: Identify if a random variable is discrete or continuous. There
are rules for a discrete distribution: probabilities of outcomes are between
zero and one, and the sum of all the probability values is one. Find the
mean and variance of a discrete probability distribution.
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