Probability Distributions: Exercises and Examples for High School Students, Schemes and Mind Maps of Mathematics

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Statistics and

Probability

Quarter 3 – Module 1:

Random Variables and

Probability Distributions

Statistics and

Probability

Quarter 3 – Module 1:

Random Variables and

Probability Distributions

What I Know

DIRECTION: Write your answer on a separate sheet of paper.

A. Read the statements carefully and choose the letter of the best

answer.

1. If two coins are tossed once, which is NOT a possible value of the random variable for the number of heads? A. 0 B. 1 C. 2 D. 3
2. Which of the following is a discrete random variable? A. Length of wire ropes B. Number of soldiers in the troop C. Amount of paint used in repainting the building D. Voltage of car batteries
3. Which formula gives the probability distribution shown by the table? X 3 4 5 P(X) 1/3 1/4 1/ A. P(X) = X B. P(X) = 1/X C. P(X) = X/ D. P(X) = X/
4. How many ways are there in tossing two coins once? A. 4 B. 3 C. 2 D. 1
5. It is a numerical quantity that is assigned to the outcome of an experiment. A. random variable B. variable C. probability D. probability distribution

B. Classify the following random variables as discrete or continuous****.

1. The weight of the professional wrestlers
2. The number of winners in lotto for each day
3. The area of lots in an exclusive subdivision
4. The speed of a car
5. The number of dropouts in a school per district C. Determine the values of the random variables in each of the following distributions.
6. Two coins are tossed. Let T be the number of tails that occur. Determine the values of the random variable T.
7. A meeting of envoys was attended by 4 Koreans and 2 Filipinos. If three envoys were selected at random one after the other, determine the values of the random variable F representing the number of Filipinos.

What’s New

The answer in this question requires an understanding of random variables. You can do it! Aja!

Definitions of Random Variable

A random variable is a result of chance event, that you can measure or

count.

A random variable is a numerical quantity that is assigned to the

outcome of an experiment. It is a variable that assumes numerical

values associated with the events of an experiment.

A random variable is a quantitative variable which values depends on

change.

NOTE:

We use capital letters to represent a random variable. Mary Ann, Hazel, and Analyn want to know what numbers can be assigned for the frequency of heads that will occur in tossing three coins. Can you help them? Thanks! Formatted: Font: 12 pt Formatted: Font: 12 pt Formatted: Font: 12 pt Formatted: Font: Bold, Italic

Example 1 Suppose two coins are tossed and we are interested to determine the number of tails that will come out. Let us use T to represent the number of tails that will come out. Determine the values of the random variable T. Solution: Steps Solution

1. List the sample space S = {HH, HT, TH, TT}
2. Count the number of tails in each outcome and assign this number to this outcome. Outcome Number of Tails (Value of T) HH 0 HT 1 TH 1 TT 2
3. Conclusion The values of the random variable T (number of tails) in this experiment are 0, 1 and 2. Example 2 Two balls are drawn in succession without replacement from an urn containing 5 orange balls and 6 violet balls. Let V be the random variable representing the number of violet balls. Find the values of the random variable V. Solution: Steps Solution

1. List the sample space S = {OO, OV, VO, VV}

Example 4 Four coins are tossed. Let T be the random variable representing the number of tails that occur. Find the values of the random variable T. Solution: Steps Solution

1. List the sample space S = {HHHH, HHHT, HHTH, HHTT, HTHH,

HTHT, HTTH, HTTT, THHH, THHT, THTH,

THTT, TTHH, TTHT, TTTH, TTTT}

2. Count the number of tails in each outcome and assign this number to this outcome. Outcome Number of tails (Value of T) HHHH 0 HHHT 1 HHTH 1 HHTT 2 HTHH 1 HTHT 2 HTTH 2 HTTT 3 THHH 1 THHT 2 THTH 2 THTT 3 TTHH 2 TTHT 3 TTTH 3 TTTT 4
3. Conclusion The values of the random variable T (number of tails) in this experiment are 0, 1, 2, 3, and 4.

Example 5 A pair of dice is rolled. Let X be the random variable representing the sum of the number of dots on the top faces. Find the values of the random variable X. Solution: Steps Solution

1. List the sample space S =

2. Count the sum of the number of dots in each outcome and assign this number to this outcome. Outcome Sum of the number of dots (Value of X) (1, 1) 2 (1, 2), (2, 1) 3 (1, 3), (3, 1), (2, 2) 4 (1, 4), (4, 1), (2, 3), (3, 2)

2. Count the number of tails in each outcome and assign this number to this outcome. The values of the random variable T (number of tails) in this experiment are 0, 1, and 2. Outcome Number of Tails (Value of T) HH 0 HT 1 TH 1 TT 2
3. Construct the frequency distribution of the values of the random variable T. Number of Tails (Value of T) Number of Occurrence (Frequency) 0 1 1 2 2 1 Total 4
4. Construct the probability distribution of the random variable T by getting the probability of occurrence of each value of the random variable. The probability distribution of the random variable T can be written as follows: T 2 1 0 P(T) 1/4 1/2 1/ Number of Tails (Value of T) Number of Occurrence (Frequency) Probability P(T) 0 1 1/ 1 2 2/4 or 1/ 2 1 1/ Total 4 1

5. Construct the probability histogram. Using Example 2 in the previous page, Steps Solution

1. List the sample space S = {OO, OV, VO, VV}

2. Count the number of violet balls in each outcome and assign this number to this outcome. The values of the random variable V (number of violet balls) in this experiment are 0, 1, and 2. Outcome Number of Violet Balls (Value of V) OO 0 OV 1 VO 1 VV 2
3. Construct the frequency distribution of the values of the random variable V. Number of Violet Balls (Value of V) Number of Occurrence (Frequency) 0 1 1 2 2 1

T

P(T)

2. Count the number of tails in each outcome and assign this number to this outcome. The values of the random variable T (number of tails) in this experiment are 0, 1, 2, 3, and 4. Outcome Number of tails (Value of T) HHHH 0 HHHT 1 HHTH 1 HHTT 2 HTHH 1 HTHT 2 HTTH 2 HTTT 3 THHH 1 THHT 2 THTH 2 THTT 3 TTHH 2 TTHT 3 TTTH 3 TTTT 4
3. Construct the frequency distribution of the values of the random variable T. Number of Tails (Value of T) Number of Occurrence (Frequency) 0 1 1 4 2 6 3 4 4 1 Total 16

4. Construct the probability distribution of the random variable T by getting the probability of occurrence of each value of the random variable. The probability distribution of the random variable T can be written as follows: T 0 1 2 3 4 P(T) 1/16 1/4 3/8 1/4 1/ Number of Tails (Value of T) Number of Occurrence (Frequency) Probability P(T) 0 1 1/ 1 4 4/16 or 1/ 2 6 6/16 or 3/ 3 4 4/16 or 1/ 4 1 1/ Total 16 1

2. Count the sum of the number of dots in each outcome and assign this number to this outcome. The values of the random variable X (sum of the number of dots) in this experiment are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. Outcome Sum of the number of dots (Value of X) (1, 1) 2 (1, 2), (2, 1) 3 (1, 3), (3, 1), (2, 2) 4 (1, 4), (4, 1), (2, 3), (3, 2) 5 (1, 5), (5, 1), (2, 4), (4, 2), (3, 3) 6 (1, 6), (6, 1), (2, 5), (5, 2), (4, 3), (3, 4)

3. Construct the frequency distribution of the values of the random variable X. Sum of the number of dots (Value of X) Number of Occurrence (Frequency) 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4 10 3 11 2

Total 36

4. Construct the probability distribution of the random variable X by getting the probability of occurrence of each value of the random variable. The probability distribution of the random variable X can be written as follows: X 2 3 4 5 6 7 8 9 10 11 12

P(X) 1

Sum of the number of dots (Value of X) Number of Occurrence (Frequency) Probability P(X) 2 1 1/ 36 3 2 2/36 or 1/ 4 3 3 / 3 6 or 1/ 5 4 4/ 3 6 or 1/ 9 6 5 5/ 7 6 6/36 or 1/ 8 5 5/ 9 4 4/36 or 1/ 10 3 3/36 or 1/ 11 2 2/36 or 1/ 12 1 1/ 36 Total 36 1