Mr. Mark Anthony Garcia, M.S.
Mathematics Department
De La Salle University
Situation: Random Variable
Consider the experiment of tossing three
coins.
 Then the sample space S will contain
the elements HHH, HHT, HTH, THH,
TTH, THT, HTT and TTT.
 Let X be a variable representing the
number of heads that occur in each
outcome.

Situation: Random Variable
The following table shows the assignment
of values to each outcome.
Outcome
HHH
HHT
HTH
THH
X
3
2
2
2
Outcome
TTT
TTH
THT
HTT
X
0
1
1
1
Situation: Random Variable
The only possible values for X are 0, 1,
2 and 3.
 The variable X is viewed as a random
variable.

Random Variable
A random variable is a function that
associates a real number with each
element in the sample space.
Probability Distribution
A probability distribution shows the
probability of each value x of the random
variable X.
Example 1: Probability
Distribution
Consider the experiment of tossing three
coins and let X be the random variable
representing the number of heads that
occur in each outcome.
Outcome
HHH
HHT
HTH
THH
X
3
2
2
2
Outcome
TTT
TTH
THT
HTT
X
0
1
1
1
Example 1: Probability
Distribution
The probability distribution of X is given in
the following table.
x
P(X=x)
0
1/8
1
3/8
2
3/8
3
1/8
Example 1: Probability
Distribution
The probability that there is no head in
the outcome (TTT) is P(X=0) = 1/8.
 The probability that there is exactly one
head in the outcome (HTT, THT, TTH) is
P(X=1) = 3/8.

Example 1: Probability
Distribution
The probability that there is exactly two
heads in the outcome (HHT, HTH, THH)
is P(X=2) = 3/8.
 The
probability that the outcome
contains all heads (HHH) is P(X=3) =
1/8.

Properties of Probability
Distribution
1.
2.
Each probability P(X=x) is greater than
0.
The sum of all probabilities is equal to
1.
Example 2: Probability
Distribution
Consider the experiment of rolling a pair
of dice.
 Let X be the random representing the
sum of the top faces of the pair of dice.

Example 2: Probability
Distribution
The following table shows the probability
distribution for X.
x
2
3
4
5
P(X=x)
1/36
2/36
3/36
4/36
x
6
7
8
9
P(X=x)
5/36
6/36
5/36
4/36
x
10
11
12
P(X=x)
3/36
2/36
1/36
Situation
Suppose that two coins are tossed 16
times. The following table gives the result:
HT
TT
HH
TH
HH
HT
HH
TT
TT
TH
HH
HT
TH
TT
HT
TT
Situation
Let X be the random variable representing
the number heads per toss.
HT
TT
HH
TH
1
0
2
1
HH
HT
HH
TT
2
1
2
0
TT
TH
HH
HT
0
1
2
1
TH
TT
HT
HH
1
0
1
2
Situation

What is the average number of heads
per toss of the two coins?
1+0+2+1+2+1+2+0+0+1+2+1+1+0+1+2

16
17

= 1.0625
16
Situation
We may view this average as the
mathematical expectation.
 This means that for every toss of two
coins, we expect that there would be 1
head in the outcome.
 However,
this result is from an
experiment.

Situation


Moreover, we can write the solution in
the following form.
0
4
16
+1
x
P(X=x)
7
16
+2
0
4/16
5
16
=
17
16
1
7/16
= 1.0625
2
5/16
Mathematical Expectation
Let X be a random variable with probability
distribution 𝑃(𝑋 = 𝑥) . The mean or
expected value of X is given by 𝐸 𝑋
= [𝑥 ∙ 𝑓 𝑥 ].
Example 3: Expectation
Suppose that in a game of tossing two
coins, you win PhP100.00 when two heads
come out and you lose Php50.00
otherwise. What is your expected gain?
Example 3: Expectation
Let X be the random variable
representing the player’s gain per toss.
 Player wins Php100 when HH comes
out.
 Otherwise, player loses Php50 when HT,
TH and TT come out

Example 3: Expectation
The probability distribution table is
shown below.
 Using mathematical expectation, we
1
3
have
𝐸 𝑋 = 100
+ −50

4
= −𝑃ℎ𝑝12.50.
x
P(X=x)
Php100
1/4
4
-Php50
3/4
Example 4: Expectation
Suppose that in a game, you win
PhP50.00 when you draw a heart card and
you lose PhP30.00 otherwise. What is
your expected gain?
x
P(X=x)
Php50
13/52
-Php30
39/52
Example 4: Expectation

The expected gain (loss) of the player is
13
39
given by 𝐸 𝑋 = 50
+ (−30)
.
52

So, 𝐸 𝑋 = −𝑃ℎ𝑝10.
52
Example 5: Expectation
By investing in a particular stock, a
person can make a profit in one year of
$4000 with probability 0.3 or take a loss
of $1000 with probability 0.7. What is
this person’s expected gain?
 𝐸 𝑋 = 4000 0.3 + (−1000)(0.7)
 𝐸 𝑋 = $500

Exercises
1.
In a gambling game, a man is paid $5 if
he gets all heads or all tails when three
coins are tossed, and he will pay out $3
if either one or two heads show. What is
his expected gain?
Exercises
2.
In a gambling game, a woman is paid
$3 if she draws a jack or a queen and
$5 if she draws a king or an ace from
an ordinary deck of 52 playing cards. If
she draws any other card, she loses.
How much should she pay to play if the
game is fair?
Exercises
3.
A game in a TV show has two parts. In
the first part, the contestants will play
an elimination game and the winner will
gain 50,000 pesos. The winner will
have a chance to spin a spinner with 8
possible winnings: return 50,000 pesos,
divide 50,000 by 2, win 10,000, win
25,000, win 50,000, win 100,000, win
250,000 and win 500,000.
Exercises
A.
Let X represents the random variable
representing the total winnings of the
contestant. Construct a probability
distribution table X.
B.
What is the expected amount of winnings of
the contestant?