STATISTICS AND
PROBABILITY
Google Meeting 1.1
Grade 11
Mrs. Mary Ann C. Demandante
STATISTICS AND
PROBABILITY
Random Variables and
Probability Distribution
SESSION AGREEMENT
Everyone must
be OPEN CAM
Everyone is
throughout the
responsible for
session.
the success of
this session. Be
fully present.
Everyone’s
input and
questions
should be
equally valued.
Respect the
person talking;
don’t take part
in side
conversations.
Listen & ask
meaningful
questions.
Keep your
audio muted
during sessions.
Discussions,
questions, and
critiques will
focus on ideas
and issues, not
on people.
All are
expected to
participate in &
attend all
sessions. We
hope nobody
left the session.
Keep an open
mind and listen
constructively.
Encourage
others, respect
differences, and
work together.
3
STATISTICS AND PROBABILITY
WEEK 1
WHAT I NEED TO KNOW
• After going through this session you are expected
to:
1. illustrate random variables and find its possible
values
2. differentiate discrete and continuous variables
3. illustrate a probability distribution for a discrete
random variable and its properties
4. compute probabilities corresponding to
a given random variable
5. construct the probability mass function
of a discrete random variable and its
corresponding histogram
WHAT I KNOW
LETS FIND OUT HOW FAR YOU MIGHT
ALREADY KNOW ABOUT THIS TOPIC.
PLEASE TAKE THIS CHALLENGE! HAVE FUN!
ACTIVITY: “ILISTA MO, TITINGNAN KO”
Directions: Learners will provide sheet of paper and
ballpen. Questions will be flashed on the screen and
learners will write their answers on the piece of
paper and will be check right after the questions.
Correct answer will be rewarded with 1 point.
DIRECTIONS: LIST DOWN THE SAMPLE
SPACE FOR THE GIVEN EXPERIMENT
1. Tossing a die
Answer: {1, 2, 3, 4, 5, 6}
2. Tossing 2 coins
Answer: {HH, HT, TT, TH}
3. Tossing a coin and a die
Answer: {1H, 2H, 3H, 4H, 5H, 6H,1T, 2T, 3T, 4T,
5T, 6T}
4. Tossing three coins
Answer:{HHH, HHT, HTH, THH, TTT, TTH, THT, HTT}
5. Drawing a jack from standard deck of cards
Answer:{J of heart, J of diamond, J of spade, J of club}
6. Tossing a pair of dice
Answer:{(1,1), (1,2), (1,3), (1,4), (1,5), (1,6),…, (6,6)} –
36 outcomes
7. Drawing a card from standard deck of cards
Answer:{}
8. Spinning a wheel
Answer:{1, 2, 3, 4, 5, 6, 7, 8}
9.
Choosing a vowel from the alphabet
Answer:{a, e, i, o, u}
10. Getting a sum of 7 when a pair of dice is tossed
Answer:{(1,6), (6,1), (2,5), (5,2), (3,4), (4,3)}
WHAT’S IN!
RANDOM VARIABLES
Random Variable
Continuous
Discrete
Mean
Variance
Standard
Deviation
A random variable is a variable whose possible
values are determined by chance.
A random variable is typically represented by an
uppercase letter, usually X, while its
corresponding lowercase letter, x, is used to
represent one of its values.
• Example: A coin is tossed thrice. Let
the variable X represent the number
of heads that results from this
experiment.
Number of Heads (X)
3
2
2
1
2
1
1
0
The value of the variable X can be 0, 1, 2, or 3. Then in this example, X is
a random variable.
WHAT’S NEW!
• A discrete random variable can only take a finite
(countable) number of distinct values. Distinct
values mean values that are exact and can be
represented by nonnegative whole numbers.
A continuous random variable can assume an infinite
number of values in an interval between two
specific values. This means they can assume values
that can be represented not only by nonnegative
whole numbers but also by fractions and decimals.
These values are often results of measurement.
Determine if the random variable X or Y is discrete or continuous:
1.
X = number of points scored in the last season by a randomly selected
basketball player in the PBA
Discrete
2.
Y = the heights of a randomly selected student inside the library in cm
Continuous
3.
X = number of birds in a nest
4.
Y = the hourly temperature last
Sunday
Discrete
Continuous
WHAT IS IT
1. What is the probability of getting a tail in
flipping a coin?
1/2
2. What is the probability of getting a heart in a
single draw from a deck of card? 13/52
3. What is the probability of getting an even
3/6 or 1/2
number in rolling a die?
QUESTIONS:
4. What is the probability of getting a face card in a
single draw from a deck of card? 12/52
5. What is the probability of getting a sum of six
when a pair of dice is tossed? 5/36
WHAT’S MORE
Probability Distribution of Discrete
Random Variable
A listing of all possible values of a discrete random
variable along with their corresponding
probabilities is called a discrete probability
distribution (probability mass function).
The discrete probability distribution can be
presented in tabular, graphical, or formula
form.
PROPERTIES OF DISCRETE
PROBABILITY DISTRIBUTION
•
Example 1: Determine whether each of the following is a
discrete probability distribution:
a.
x
1
2
3
4
5
P(x)
0.10
0.20
0.25
0.40
0.05
Yes. The probability of each value of a discrete random
variable is between 0 and 1 and the sum of all probabilities
is 1.
b.
x
1
2
3
4
5
P(x)
0.05
0.25
0.33
0.25
0.08
No. Although the probability of each value is between 0 and
1, the sum of their probabilities is not equal to 1.
•
x
1
2
3
4
P(x)
0.21
29c
0.29
0.21
Example 3: Suppose three coins are tossed. Let X
be the number of tails that occur. Find the
probability of each of the values of the random
variable X.
SS = {TTT, TTH, THT, HTT, HHT, HTH, THH, HHH}
P(0) = 1/8, P(1) = 3/8, P(2) = 3/8, P(3) = 1/8
The probability distribution or the probability mass
functionx of discrete
variable
X
0
1
2
3
P(x)
1/8
3/8
3/8
1/8
Example 4: The spinner below is divided into 12 sections. Let X
be the score where the arrow will stop (numbered as 1, 2, 3, 4,
5).
a.
Find the probability that the arrow will stop at 1, 2, 3, 4, and
5.
b. Construct the discrete probability distribution of the
random variable X and its corresponding histogram.
a.
P(1) = 1/12
P(2) = 2/12 = 1/6
P(3) = 3/12 = ¼
P(4) = 2/12 = 1/6
P(5) = 4/12 = 1/3
b. The discrete probability distribution is:
x
1
2
3
4
5
P(x)
1/12
1/6
1/4
1/6
1/3
Histogram:
P(x)
5/12
4/12
3/12
2/12
1/12
x
1
2
3
4
5
6
RANDOM VARIABLES AND
PROBABILITY DISTRIBUTION
A. Classify the following random variables as discrete or
continuous
1. X = the number of mobile phones sold in one week in AB
store
2.Y = the weight in pounds of newly born babies in a hospital
nursery
3. X = the number of cars in a parking lot every noon
4.Y = the number of gifts received by a birthday celebrator
5. X = the length of time spent in playing video games in minutes
ACTIVITY MODE!
Refer to the worksheets and answer
Exercises A - E.
CLOSURE
MEAN AND VARIANCE OF A
DISCRETE RANDOM VARIABLE
The mean of a discrete random variable X is also
called the expected value of X. It is the weighted
average of all the values that the random variable X
would assume in the long run. The discrete random
variable X assumes values or outcomes in every trial
of an experiment with their corresponding
probabilities
MEAN AND VARIANCE OF A
DISCRETE RANDOM VARIABLE
The expected value of x is the average
of the outcomes that is likely to be
obtained if the trials are repeated over
and over again.
•
Example 1: A researcher surveyed the households in a small town. The
random variable X represents the number of college graduates in the
households.The probability distribution of X is shown below:
Find the mean or expected value of X.
X
0
1
2
P(x)
0.25
0.50
0.25
Solution:
x
P(x)
x • P(x)
0
0.25
0
1
0.50
0.50
2
0.25
0.50
The expected value is 1. So the average number of college graduates in the
household of the small town is 1.
Example 2: The probabilities that a costumer will buy 1, 2, 3,
4, or 5 items in a grocery store are 3/10, 1/10, 1/10, 2/10,
and 3/10. What is the average number of items that a
customer will buy?
Solution:
x
P(x)
x • P(x)
1
3/10
3/10
2
1/10
2/10
3
1/10
3/10
4
2/10
8/10
5
3/10
15/10
The mean is 3.1. This implies that the average number of
items that the customer will buy is 3.1.
•
•
x
0
1
2
3
4
P(x)
0.1
0.2
0.3
0.3
0.1
•
x
P(x)
x•P(x)
0
0.1
0
0-2.1 = -2.1
4.41
0.441
1
0.2
0.2
1-2.1 = -1.1
1.21
0.242
2
0.3
0.6
2-2.1 = -0.1
0.01
0.003
3
0.3
0.9
3-2.1 = 0.9
0.81
0.243
4
0.1
0.4
4-2.1 = 1.9
3.61
0.361
•
•
x
P(x)
x•P(x)
0
0.1
0
0
0
1
0.2
0.2
1
0.2
2
0.3
0.6
4
1.2
3
0.3
0.9
9
2.7
4
0.1
0.4
16
1.6
Exercises
Refer to the worksheets and answer exercises F - G.
PROBLEMS INVOLVING MEAN AND
VARIANCE OF PROBABILITY
DISTRIBUTION
Example 1: The officers of SJA Class 71 decided to conduct a
lottery for the benefit of the less privileged students of their
alma mater. Two hundred tickets will be sold. One ticket will
win P5,000 price and the other tickets will win nothing. If you
buy one ticket, what will be your expected gain?
•
x
P(x)
x•P(x)
0
0.995
0
5,000
0,005
25
•
x
P(x)
x•P(x)
-30
0.5
-15
450
50
0.5
25
1250
Exercises
Refer to the worksheets and answer exercises H.
CLOSURE