AIOU 1429 Solved Assignment 1 Spring 2025

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AIOU 1429 Business Mathematics Solved Assignment 1 Spring 2025

AIOU 1429 Assignment 1

Q1.(a There are 16 green, 20 red and 24 yellow balls in a basket. If we pick a ball at random what is the probability that
i. The ball is green.
ii. The ball is not green and red. (20 Marks)

Step 1: Total Number of Balls

There are:

  1. 16 green balls
  2. 20 red balls
  3. 24 yellow balls

The total number of balls is 16 + 20 + 24 = 60.

Step 2: Calculating the Probabilities

i. Probability that the ball is green

  1. The number of green balls is 16.
  2. Total number of balls is 60.
  3. The probability of picking a green ball is:
    P(green) = 16 / 60, which simplifies to 4/15.

ii. Probability that the ball is not green and red

  1. This phrase means the ball is neither green nor red.
  2. Therefore, the ball must be yellow.
  3. The number of yellow balls is 24.
    The probability of picking a yellow ball is:
    P(yellow) = 24 / 60, which simplifies to 2/5.

Final Answers:

  1. The probability that the ball is green is 4/15.
  2. The probability that the ball is not green and red (i.e., the ball is yellow) is 2/5.

Q1.(b Differentiate between continuous and discrete random variables with the help of examples.

Continuous Random Variables

A continuous random variable can take any value within a given range. It represents measurements and can have decimals or fractions.

Examples:

  1. Height of students in a class (could be 160 cm, 160.2 cm, 160.35 cm, etc.)
  2. Time taken to finish a race (could be 12.45 seconds, 12.467 seconds, etc.)
  3. Temperature in a city (e.g., 24.5°C, 24.51°C)

Since these values can be infinitely precise, they require intervals rather than exact numbers.

Discrete Random Variables

A discrete random variable takes specific, countable values. It represents things that are counted rather than measured.

Examples:

  1. Number of students in a class (can be 30, 31, 32—not 30.5!)
  2. Number of cars in a parking lot (e.g., 15, 20, 25—not fractions)
  3. Number of goals scored in a soccer match (e.g., 3 goals, 5 goals—never 3.7 goals!)

Since discrete variables come in distinct numbers, they can be listed or counted directly.

Summary of Key Differences

Feature Continuous Random Variable Discrete Random Variable
Values Can take any value within an interval Specific, countable numbers
Nature Measured (e.g., height, weight, temperature) Counted (e.g., number of students, goals, cars)
Examples Time, speed, distance Coins flipped, books on a shelf

Q2. The number of fire alarms pulled each hour fluctuates in Islamabad. The probability table of different alarms per hour is shown(20 Marks)

No. of Alarm Pulled Probability
Less than 8 0.12
8 0.24
9 0.28
10 0.26
More than 10 0.10

a) Probability that more than 8 alarms will be pulled

  1. The probability of pulling 9 alarms is 0.28.
  2. The probability of pulling 10 alarms is 0.26.
  3. The probability of pulling more than 10 alarms is 0.10.
  4. The total probability is:
    0.28 + 0.26 + 0.10 = 0.64.

b) Probability that the number of alarms pulled is between 8 and 9 (inclusive)

  1. The probability of pulling 8 alarms is 0.24.
  2. The probability of pulling 9 alarms is 0.28.
  3. The total probability is:
    0.24 + 0.28 = 0.52.

Final Answers:

  1. The probability that more than 8 alarms will be pulled is 0.64.
  2. The probability that the number of alarms pulled is between 8 and 9 (inclusive) is 0.52.

Q3. The data on ocean storms in the USA for the last fifty years is given below(20 Marks)


No. of Storms Frequency
0 2
1 7
2 10
3 15
4 7
5 12
6 7
Total: 60

a) Construct probability distribution for this data.

The probability of each event occurring is given by:

P(X) = Frequency / Total Count

Since the total number of observations is 60, we compute probabilities for each storm count:


No. of Storms (X) Frequency (f) Probability P(X)
0 2 2/60 = 0.0333
1 7 7/60 = 0.1167
2 10 10/60 = 0.1667
3 15 15/60 = 0.2500
4 7 7/60 = 0.1167
5 12 12/60 = 0.2000
6 7 7/60 = 0.1167

b) Draw a histogram for this distribution.

AIOU 1429 Solved Assignment

Q4.(a Solve the second degree equation and find the nature of its roots.
y^2-y-2=0(10 Marks)

AIOU 1429 Solved Assignment

Q4.(b Solve the inequality and represent the solution on the real line.(10 Marks)

AIOU 1429 Solved Assignment

Q5.(a Find the point of intersection of the following lines if it exists.
x + 2y = 3
2x - y = 1(10 Marks)

Let's solve for the intersection point by solving the system of equations:

x + 2y = 3

2x - y = 1

Step 1: Express One Variable in Terms of the Other

From the first equation, solve for x:

x = 3 - 2y

Step 2: Substitute in the Second Equation

Replace x in the second equation:

2(3 - 2y) - y = 1

Expanding:

6 - 4y - y = 1

6 - 5y = 1

Step 3: Solve for y

-5y = -5

y = 1

Step 4: Solve for x

Substituting y = 1 into x = 3 - 2y:

x = 3 - 2(1) = 1

Conclusion:

The lines intersect at (1,1).

Q5.(b Let C mean Celsius degree and F mean Fahrenheit temperature scale. Find a linear equation for C if its slope is 5/9 and the C-intercept is -100/9.(10 Marks)

To find the linear equation for C in terms of F, we use the standard equation of a line:

C = mF + b

where:

m is the slope (5/9)

b is the C-intercept (-100/9)

Substituting these values:

C = (5/9)
F - (100/9)

Conclusion:

The required linear equation relating Celsius (C) to Fahrenheit (F) is:

C = (5/9)
F - (100/9)

AIOU 1429 Solved Assignment 2 Spring 2025

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