SAN ROQUE COLLEGE de CEBU

LILO-AN • CORDOVA • BOGO

“For the Greater Glory of God!”

www.srcds.edu.ph

WEEKLY LEARNER’S MODULE

S.Y 2021-2022

Subject: Statistics and Probability Level: GRADE 12

Quarter: First (Midterm) Week: 3 (Three)

PROBLEMS INVOLVING MEAN AND VARIANCE OF PROBABILITY

DISTRIBUTION

I. Most Essential Learning Competency:

● Solves problems involving mean and variance of probability distributions.

II. Learning Objectives:

1. Understands problems involving mean and variance of probability distributions.
2. Computes the mean and variance of probability distributions.
3. Interprets the mean and variance of probability distributions.

III. Topic: PROBLEMS INVOLVING MEAN AND VARIANCE OF PROBABILITY

Resources: Next Century Mathematics Statistics and Probability

IV. Concept (Generalization):

● The mean of a discrete random variable can be thought of as “anticipated” value. It is the
average that is expected to be the result when a random experiment is continually repeated.
● The variance of discrete random variables is the expected value of the square of the difference
between the assumed value of random variable and the mean.
● A problem solving involving probability distributions is very essential when dealing with real-life
a problem that involves chances.
V. Discussion:

The mean of a discrete random variable can be thought of as “anticipated value”. It is the sum of the
possible outcomes of the experiment multiplied by their corresponding probabilities.
Just like in the previous topic, the mean will be called expected value.

Example 1 The officer of SJA class 71 decide to conduct a lottery for the benefit of the less privilege students
of their alma matter. Two hundred tickets will be sold. One ticket will win ₱5,000.00 price and the
other ticket will win nothing. If you will buy one ticket, what will be your expected gain?

Solution: One ticket will have a gain of ₱5,000.00 but the probability of winning will only be or 0.005. The
remaining tickets will have a gain of ₱0.00, and the probability will be or 0.995

( ) ( )
0 0.995 0
5000 0.005 25

∑ [ ( )]
 Page |2

( ) ∑[ ( )] , the expected gain is ₱25.00

Example 2 The officer of the faculty club of a public high school are planning to sell 160 tickets to be
raffled during the Christmas party. One ticket will win ₱3,000.00. The other ticket will win
nothing. If you are a faculty member of the school and you will buy one ticket, what will be
the expected value and the variance of your gain?

Solution: One ticket will have a gain of ₱3,000.00 but the probability of winning will only be or 0.00625.
The remaining tickets will have a gain of ₱0.00 and the probability will be or 0.99375.
( ) ( ) ( ) ( ) ( )
0 0.99375 0 -18.75 351.5625 349.3652344
3000 0.00625 18.75 2981.25 8887851.563 55549.07227

∑ [ ( )] ∑ [( ) ( )] 55898.4375

( ) ∑ [ ( )]

∑ [( ) ( )]

The expected value is ₱18.75. This amount is expected gain.

The variance of you gain is 55898.44 and it indicates how spread out the values of x are around the
mean.

Example 3 Jack tosses an unbiased coin. He receives ₱50.00 if a head appears and pays ₱30.00 if a tail
appears. Find the expected value and variance of his gain.

Solution:
( ) ( ) ( ) ( ) ( )
-30 0.5 -15 -40 1600 800
50 0.5 25 40 1600 800
∑ [( ) ( )] 1600
∑ [ ( )]

In tossing an unbiased coin, there are two elementary events which are equally likely. The probability
of occurrence of each of this elementary event is 0.5. Jack will have a gain of ₱50.00 if a head appears and a
loss of ₱30 if a tail appears. But then, the probability of a head appearing and the probability of the tail
appearing are the same. The expected gain is ₱10. The variance is 1600.
 Page |3

SAN ROQUE COLLEGE de CEBU

LILO-AN • CORDOVA • BOGO

“For the Greater Glory of God!”

www.srcds.edu.ph

WEEKLY LEARNER’S MODULE

S.Y 2021-2022

Subject: Statistics and Probability Level: GRADE 12

Quarter: First (Midterm) Week: 3 (Three)
PROBLEMS INVOLVING MEAN AND VARIANCE OF PROBABILITY
DISTRIBUTION

Name:__________________________________________ Section:_____________________________
Parent’s Signature: _______________________________ Date Submitted:___________________________

VI. Activities for the Week

Monday Study the week’s module
Tuesday-Saturday Do activity/activities and assessment correctly and completely

ACTIVITY 1.3
Directions: Solve what is asked in the following problems below. Show your solution.

1. The officers of the Science Club are planning to sell 125 tickets to be raffled during the school’s
foundation day. One ticket will win ₱2000.00 and the other tickets will win nothing. If you will buy one
ticket, what will be your expected gain?, what will be the variance?

( ) ( ) ( ) ( ) ( )

∑ [ ( )] ∑ [( ) ( )]

2. A lottery will be conducted for the benefit of the poor but deserving student of a certain school. Four
hundred tickets will be sold. One ticket will win ₱2000.00 and the other tickets will win nothing. If you
will buy one ticket, what will be your expected gain? , what will be the variance?
( ) ( ) ( ) ( ) ( )

∑ [ ( )] ∑ [( ) ( )]

3. Laverny tosses an unbiased coin. He receives ₱100.00 if a head appears and he pays ₱40.00 if a tail
appears. Find the expected value and the variance of his gain.

( ) ( ) ( ) ( ) ( )

∑ [ ( )] ∑ [( ) ( )]
 Page |4

SAN ROQUE COLLEGE de CEBU

LILO-AN • CORDOVA • BOGO

“For the Greater Glory of God!”

www.srcds.edu.ph
WEEKLY LEARNER’S MODULE

S.Y 2021-2022

Subject: Statistics and Probability Level: GRADE 12

Quarter: First (Midterm) Week: 3 (Three)
PROBLEMS INVOLVING MEAN AND VARIANCE OF PROBABILITY
DISTRIBUTION
Name:__________________________________________ Grade & Sec.:_____________________________
Parent’s Signature: _______________________________ Date Submitted:___________________________

VII. Assessment

Directions: Solve what is asked in the following problems below. Show your solution.

1) Five hundred tickets will be sold and these will be raffled during the town fiesta. One of these
tickets will win ₱3000.00 and the rest will win nothing. What will be the expected outcome and
variance of your gain if you will buy one of the tickets?
a.
( ) ( ) ( ) ( ) ( )

b. ∑ [ ( )]

c. ∑ [( ) ( )]

2) Five hundred tickets will be sold and these will be raffled during the town fiesta. One of these
tickets will win ₱3000.00 and the rest will win nothing. What will be the expected outcome and
variance of your gain if you will buy one of the tickets?
a.

( ) ( ) ( ) ( ) ( )

b. ∑ [ ( )]

c. ∑ [( ) ( )]
 Page |5

3) You are in a carnival and you see a game. The rule says that the outcome in the game is a
random variable from 1 to 14 and that if the outcome is even you win ₱50. If the outcome is
odd, you win nothing. If you play the game, what will be the expected outcome and variance of
your gain

a.
( ) ( ) ( ) ( ) ( )

b. ∑ [ ( )]

c. ∑ [( ) ( )]

4. Jack tosses an unbiased coin. He receives ₱50.00 if a head appears and he pays ₱60.00 if a tail appears.
Find the expected value and the variance of his gain.

a.
( ) ( ) ( ) ( ) ( )

b. ∑ [ ( )]

c. ∑ [( ) ( )]

Prepared by:

ULYSIS L. PEVIDA
SUBJECT TEACHER