UNIT II Statistical Probability

Statistical Probability Distributions for Civil Engineering Applications

Statistical probability distributions –Students’ t, Chi – square and F – distributions – applications

1. Introduction to Probability Distributions

In statistics, a probability distribution describes how the values of a random variable are distributed. It tells us which values a variable is likely to take and how often.

Why Probability Distributions Matter in Civil Engineering?

Civil engineers deal with real-world data such as soil strength, traffic flow, and concrete quality. Since exact values are unpredictable, we use probability distributions to make decisions.

The three important probability distributions used in Civil Engineering are:

• Student’s t-distribution (for small sample sizes)
• Chi-square distribution (for variance and independence testing)
• F-distribution (for comparing two variances)

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2. Student’s t-Distribution

Definition

The Student’s t-distribution is used when the sample size is small ($n < 30$), and the population standard deviation ($\sigma$) is unknown. It helps in estimating the true mean of a population from a small sample.

Formula

$$t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$$

where:
$\bar{x}$ = Sample mean
$\mu$ = Population mean
$s$ = Sample standard deviation
$n$ = Sample size

Characteristics of t-Distribution

• Bell-shaped and symmetric, like the normal distribution.
• More spread out than the normal distribution (fatter tails).
• As $n$ increases, the t-distribution approaches a normal distribution.

Example Problem 1

A Civil Engineering lab tests the compressive strength of concrete samples ($n = 10$). The sample mean is 40 MPa, and the standard deviation is 5 MPa. Find the t-value if the assumed true mean is 42 MPa.

Solution

Given:
$\bar{x} = 40$, $\mu = 42$, $s = 5$, $n = 10$

$$t = \frac{40 - 42}{5 / \sqrt{10}} = \frac{-2}{5 / 3.16} = \frac{-2}{1.58} = -1.27$$

So, t = -1.27

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3. Chi-Square ($\chi^2$) Distribution

Definition

The Chi-square distribution is used for testing variance and checking whether observed data fits expected data. It is used for:

1. Testing variance of a sample against a population.
2. Testing independence between two factors (e.g., rainfall and landslides).

Formula for Variance Testing

$$\chi^2 = \frac{(n-1) s^2}{\sigma^2}$$

where:
$s^2$ = Sample variance
$\sigma^2$ = Population variance
$n$ = Sample size

Characteristics of Chi-square Distribution

• Not symmetric, skewed to the right.
• Changes shape with different degrees of freedom ($df = n-1$).
• Always positive, since variance cannot be negative.

Example Problem 2

A Civil Engineer wants to check if the variance of soil moisture measurements (sample size 15) is 4%. The sample variance is 6%. Find the Chi-square value.

Solution

Given:
$s^2 = 6$, $\sigma^2 = 4$, $n = 15$

$$\chi^2 = \frac{(15-1) \times 6}{4} = \frac{14 \times 6}{4} = \frac{84}{4} = 21$$

So, $\chi^2 = 21$

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4. F-Distribution

Definition

The F-distribution is used to compare two variances to determine if they are significantly different. This is useful in material strength tests or comparing soil sample variances.

Formula

$$F = \frac{s_1^2}{s_2^2}$$

where:
$s_1^2$ = Variance of sample 1
$s_2^2$ = Variance of sample 2

Characteristics of F-Distribution

• Not symmetric, skewed to the right.
• Used only for positive values.
• Changes shape based on degrees of freedom ($df_1, df_2$).

Example Problem 3

Two concrete batches are tested for compressive strength. The variances of the first and second batches are 8 MPa² and 5 MPa² respectively. Find the F-ratio.

Solution

Given:
$s_1^2 = 8$, $s_2^2 = 5$

$$F = \frac{8}{5} = 1.6$$

So, F = 1.6

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5. Civil Engineering Applications

Student’s t-distribution Applications

• Testing material strengths (Concrete, Steel) with small sample sizes.
• Determining the mean settlement of a building foundation.

Chi-square Distribution Applications

• Checking soil quality variations.
• Analyzing traffic accident data to see if accidents depend on time of day.

F-Distribution Applications

• Comparing strength variations in different concrete mixes.
• Testing differences in earthquake intensities across different locations.

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6. Quiz Questions

1. When is Student’s t-distribution used?
2. What is the formula for Chi-square variance test?
3. Which distribution is used to compare two variances?
4. If sample size n increases, what happens to the t-distribution?
5. What is the Chi-square test used for in Civil Engineering?
6. Why do we use F-distribution instead of Chi-square sometimes?
7. In a t-test, what happens if the sample standard deviation is 0?
8. What is the minimum sample size required for a Chi-square test to be valid?
9. If two concrete samples have equal variances, what is the F-value?
10. Which test would you use to check whether traffic accidents are independent of time of day?

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This detailed lecture note covers definitions, formulas, problems, and applications of t, Chi-square, and F-distributions in Civil Engineering. 🚀

Unit II Uniform, normal, log normal, Exponential & Gamma Distributions

Lecture Notes on Probability Distributions for Continuous Random Variables

Probability distributions for continuous random variables – Uniform, normal, log normal, exponential and Gamma distributions – Applications

Introduction

Probability distributions for continuous random variables play a fundamental role in probability and statistics. In this lecture, we will explore five essential continuous distributions:

• Uniform Distribution
• Normal Distribution
• Log-normal Distribution
• Exponential Distribution
• Gamma Distribution

Each of these distributions has important applications in civil engineering, including material strength analysis, reliability assessment, and environmental studies.

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1. Uniform Distribution

Definition

A random variable $X$ follows a uniform distribution on the interval $(a, b)$ if its probability density function (PDF) is:

$$f(x) = \begin{cases} \frac{1}{b-a}, & a \leq x \leq b \\ 0, & \text{otherwise} \end{cases}$$

Cumulative Distribution Function (CDF)

$$F(x) = \begin{cases} 0, & x < a \\ \frac{x-a}{b-a}, & a \leq x \leq b \\ 1, & x > b \end{cases}$$

Mean and Variance

$$E[X] = \frac{a+b}{2}, \quad Var(X) = \frac{(b-a)^2}{12}$$

Application in Civil Engineering

• Traffic Flow Analysis: Modeling vehicle arrival times at an intersection.
• Construction Materials: Estimating the uniform spread of loads on a beam.

Example Problem

A pedestrian waits for a bus that arrives uniformly between 10:00 and 10:30 AM. What is the probability they wait less than 10 minutes?

Solution: Given $a = 0$ min and $b = 30$ min, the probability of waiting less than 10 minutes is:

$$P(X < 10) = \frac{10 - 0}{30 - 0} = \frac{10}{30} = \frac{1}{3}$$

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2. Normal Distribution

Definition

A random variable $X$ follows a normal distribution with mean $\mu$ and variance $\sigma^2$, denoted as $X \sim N(\mu, \sigma^2)$, if its PDF is:

$$f(x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}, \quad -\infty < x < \infty$$

CDF

There is no closed-form expression, but values are found using normal tables or statistical software.

Mean and Variance

$$E[X] = \mu, \quad Var(X) = \sigma^2$$

Application in Civil Engineering

• Material Strength: Concrete strength follows a normal distribution.
• Bridge Load Analysis: The distribution of vehicle weights on a bridge.

Example Problem

The compressive strength of concrete follows $N(30, 4^2)$. What is the probability that a randomly selected sample has strength above 35 MPa?

Solution: Standardizing:

$$Z = \frac{X - \mu}{\sigma} = \frac{35 - 30}{4} = \frac{5}{4} = 1.25$$

From the normal table: $P(Z > 1.25) = 1 - 0.8944 = 0.1056$. Thus, the probability is 10.56%.

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3. Log-normal Distribution

Definition

A random variable $X$ follows a log-normal distribution if $Y = \ln(X)$ follows a normal distribution.

PDF:

$$f(x) = \frac{1}{x \sigma \sqrt{2\pi}} e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}}, \quad x > 0$$

Application in Civil Engineering

• Soil Permeability: The distribution of soil permeability measurements.
• Building Service Life: Modeling the lifespan of materials subject to degradation.

Example Problem

If $Y = \ln(X)$ follows $N(2, 0.25)$, find the median of $X$.

Solution: The median of a log-normal distribution is:

$$Median(X) = e^\mu = e^2 = 7.389$$

Thus, the median is 7.39.

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4. Exponential Distribution

Definition

The exponential distribution models the time until an event occurs. The PDF is:

$$f(x) = \lambda e^{-\lambda x}, \quad x > 0$$

Mean and Variance

$$E[X] = \frac{1}{\lambda}, \quad Var(X) = \frac{1}{\lambda^2}$$

Application in Civil Engineering

• Reliability Analysis: Modeling the time between failures of construction equipment.
• Queueing Theory: Modeling the time between vehicle arrivals at toll booths.

Example Problem

A water pump fails on average once every 10 years. What is the probability it lasts more than 15 years?

Solution: Here, $\lambda = \frac{1}{10}$. The probability is:

$$P(X > 15) = e^{-\lambda x} = e^{-\frac{15}{10}} = e^{-1.5} \approx 0.2231$$

So, the probability is 22.31%.

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5. Gamma Distribution

Definition

The gamma distribution is a generalization of the exponential distribution. It is given by:

$$f(x) = \frac{\lambda^k x^{k-1} e^{-\lambda x}}{\Gamma(k)}, \quad x > 0$$

where $k$ is the shape parameter and $\lambda$ is the rate parameter.

Mean and Variance

$$E[X] = \frac{k}{\lambda}, \quad Var(X) = \frac{k}{\lambda^2}$$

Application in Civil Engineering

• Flood Risk Analysis: Modeling the time between extreme rainfall events.
• Structural Load Analysis: Modeling the cumulative load effects over time.

Example Problem

If failures in a bridge structure follow a gamma distribution with $k = 2$ and $\lambda = 1/5$, find the probability that failure occurs before 8 years.

Solution: The cumulative probability is obtained using the gamma CDF:

$$P(X \leq 8) = \int_0^8 \frac{(1/5)^2 x e^{-x/5}}{\Gamma(2)} dx$$

Using tables or software, $P(X \leq 8) \approx 0.798$, or 79.8%.

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Summary Table

Distribution	PDF	Mean	Variance	Civil Engineering Applications
Uniform	$\frac{1}{b-a}$	$\frac{a+b}{2}$	$\frac{(b-a)^2}{12}$	Traffic flow, material loads
Normal	$\frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$	$\mu$	$\sigma^2$	Concrete strength, vehicle loads
Log-normal	$\frac{1}{x \sigma \sqrt{2\pi}} e^{-\frac{(\ln x - \mu)^2}{2\sigma^2}}$	$e^{\mu + \sigma^2/2}$	$(e^{\sigma^2} -1)e^{2\mu + \sigma^2}$	Soil permeability, material lifespan
Exponential	$\lambda e^{-\lambda x}$	$1/\lambda$	$1/\lambda^2$	Equipment failure, traffic delays
Gamma	$\frac{\lambda^k x^{k-1} e^{-\lambda x}}{\Gamma(k)}$	$k/\lambda$	$k/\lambda^2$	Flood risk, structural loads

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Conclusion

Understanding these distributions helps civil engineers make informed decisions in infrastructure design, reliability analysis, and risk management.

UNIT II Assessment Questions

Assessment Questions Due : Feb 28.02.2025

Here are 2 assessment questions for each of the topics: Bernoulli, Binomial, Geometric, and Poisson distributions. These questions are practical, scenario-based, and ideal for testing understanding.

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1. Bernoulli Distribution

Question 1:
An engineer is testing the compressive strength of a concrete block. The probability that the block meets the required strength is $p = 0.8$.

• (a) What is the probability that the block fails to meet the required strength?
• (b) If the outcome is recorded as 1 for success and 0 for failure, what are the possible probabilities for $X$?

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Question 2:
A soil test is conducted to determine if the soil is fertile. The probability that the soil is fertile is $p = 0.6$.

• (a) What is the probability of finding infertile soil?
• (b) If 5 tests are conducted independently, how many tests are expected to return a positive result?

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2. Binomial Distribution

Question 1:
A batch of 10 steel rods is tested for defects. The probability that a rod is defective is $p = 0.1$.

• (a) What is the probability that exactly 3 rods are defective?
• (b) What is the probability that at least 2 rods are defective?

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Question 2:
In a quality control process, the probability of a brick being defective is $p = 0.05$. A sample of 20 bricks is inspected.

• (a) What is the probability that no brick is defective?
• (b) What is the probability that exactly 2 bricks are defective?

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3. Geometric Distribution

Question 1:
A geologist is testing rocks for mineral content. The probability of finding a rock with a high mineral content is $p = 0.3$.

• (a) What is the probability that the first high-mineral-content rock is found on the 4th test?
• (b) What is the probability that the first high-mineral-content rock is found on or before the 3rd test?

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Question 2:
A civil engineer is checking culverts to find one that meets the required specifications. The probability that a culvert meets specifications is $p = 0.4$.

• (a) What is the probability that the first suitable culvert is found on the 6th attempt?
• (b) What is the expected number of culverts the engineer needs to inspect before finding a suitable one?

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4. Poisson Distribution

Question 1:
Vehicles pass through a toll gate at an average rate of 6 per hour. The number of vehicles follows a Poisson distribution.

• (a) What is the probability that exactly 4 vehicles pass through the gate in a given hour?
• (b) What is the probability that more than 6 vehicles pass through the gate in an hour?

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Question 2:
In a rural area, landslides occur at an average rate of 3 per year. The number of landslides follows a Poisson distribution.

• (a) What is the probability of observing no landslides in a year?
• (b) What is the probability of observing at least 2 landslides in a year?

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These questions are tailored to test understanding and application of each distribution in civil engineering contexts. Let me know if you’d like solutions for any of these!

UNIT II

 Unit–II : Probability Distributions for Discrete and Continuous Random Variables Probability  distributions  for  discrete  random  variables  – Bernoulli’s,  Binomial,  Geometric and Poisson distributions – applications 

Lecture Notes: Probability Distributions for Discrete and Continuous Random Variables

Introduction to Probability Distributions

A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.

• Discrete Random Variables: These take specific, countable values (e.g., number of cars on a bridge, cracks in a concrete slab).
• Continuous Random Variables: These take any value within a range (e.g., rainfall in mm, compressive strength of concrete).

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Discrete Probability Distributions

1. Bernoulli Distribution

• Definition: A distribution for a single trial with only two outcomes: success ($p$) or failure ($1-p$).

• Formula:

  $$P(X = x) = \begin{cases} p, & \text{if } x = 1 \, (\text{success}) \\ 1-p, & \text{if } x = 0 \, (\text{failure}) \end{cases}$$

  Here, $X \in \{0, 1\}$.

• Example in Civil Engineering: A soil test is conducted to check whether it is suitable ($X=1$) or unsuitable ($X=0$) for construction.

• Numerical Example:

  • Probability that a soil sample is suitable ($p$) = 0.8.
  • Probability of being unsuitable = $1-p = 0.2$.

  Outcome probabilities:

  $$P(X=1) = 0.8, \quad P(X=0) = 0.2$$

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2. Binomial Distribution

• Definition: Models the number of successes in $n$ independent Bernoulli trials.

• Formula:

  $$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, 2, \dots, n$$

  Where:

  • $n$: Number of trials.
  • $k$: Number of successes.
  • $p$: Probability of success.

• Example in Civil Engineering: Estimating the number of defective parts in a batch of $n$ parts manufactured.

• Numerical Example:

  • Probability of a defective part ($p$) = 0.1.
  • Batch size ($n$) = 5.
  • Calculate $P(X=2)$, the probability of 2 defective parts: $$P(X=2) = \binom{5}{2} (0.1)^2 (0.9)^3 = 10 \cdot 0.01 \cdot 0.729 = 0.0729$$

  Result: Probability of 2 defective parts is $7.29\%$.

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3. Geometric Distribution

• Definition: Models the number of trials needed to get the first success.

• Formula:

  $$P(X = k) = (1-p)^{k-1} p, \quad k = 1, 2, 3, \dots$$

  Where:

  • $k$: Number of trials.
  • $p$: Probability of success.

• Example in Civil Engineering: Number of soil tests required to find the first suitable site for construction.

• Numerical Example:

  • Probability that a site is suitable ($p$) = 0.3.
  • Find $P(X=3)$, the probability that the 3rd test is the first success: $$P(X=3) = (1-0.3)^2 \cdot 0.3 = 0.7^2 \cdot 0.3 = 0.147$$

  Result: Probability is $14.7\%$.

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4. Poisson Distribution

• Definition: Models the number of events occurring in a fixed interval of time or space.

• Formula:

  $$P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k = 0, 1, 2, \dots$$

  Where:

  • $\lambda$: Average rate of occurrence.
  • $k$: Number of events.

• Example in Civil Engineering: Number of vehicles crossing a bridge in an hour.

• Numerical Example:

  • Average number of vehicles per hour ($\lambda$) = 5.
  • Find $P(X=3)$, the probability of 3 vehicles crossing: $$P(X=3) = \frac{5^3 e^{-5}}{3!} = \frac{125 \cdot 0.0067}{6} \approx 0.14$$

  Result: Probability is $14\%$.

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Continuous Probability Distributions

1. Uniform Distribution

• Definition: Models variables uniformly distributed over an interval $[a, b]$.

• Formula:

  $$f(x) = \begin{cases} \frac{1}{b-a}, & a \leq x \leq b \\ 0, & \text{otherwise} \end{cases}$$

• Example in Civil Engineering: The thickness of a concrete slab uniformly varying between 10 cm and 20 cm.

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2. Normal Distribution

• Definition: Models natural phenomena, with a symmetric bell-shaped curve.

• Formula:

  $$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

  Where:

  • $\mu$: Mean.
  • $\sigma^2$: Variance.

• Example in Civil Engineering: Variations in compressive strength of concrete.

• Numerical Example:

  • Compressive strength ($X$) is normally distributed with $\mu = 40$ MPa and $\sigma = 5$ MPa.
  • Find $P(35 \leq X \leq 45)$:

    • Standardize $X$ to $Z$:

      $$Z = \frac{X - \mu}{\sigma}$$

      For $X=35$, $Z = \frac{35-40}{5} = -1$.
      For $X=45$, $Z = \frac{45-40}{5} = 1$.

    • Use standard normal tables:

      $$P(-1 \leq Z \leq 1) = P(Z \leq 1) - P(Z \leq -1)$$ $$P(Z \leq 1) = 0.8413, \quad P(Z \leq -1) = 0.1587$$ $$P(-1 \leq Z \leq 1) = 0.8413 - 0.1587 = 0.6826$$

  Result: Probability is $68.26\%$.

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Applications in Civil Engineering

1. Bernoulli Distribution: Checking soil suitability in a binary pass/fail test.
2. Binomial Distribution: Predicting the number of defective items in a batch.
3. Geometric Distribution: Determining the number of tests to find a viable construction site.
4. Poisson Distribution: Estimating vehicle counts or defects over a time period.
5. Normal Distribution: Modeling variability in material properties like tensile strength or elasticity.

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Conclusion

Understanding probability distributions is critical in civil engineering for risk assessment, design, and decision-making under uncertainty. These distributions allow engineers to quantify variability and make informed decisions.

UNIT 1 Conditional Probability

 

Lecture Notes: Conditional Probability

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Definition

Conditional probability measures the likelihood of an event occurring given that another event has already occurred. It helps analyze dependencies between events.

Formula

If $A$ and $B$ are two events, and $P(B) > 0$, the conditional probability of $A$ given $B$ is:

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

Where:

• $P(A|B)$: Probability of $A$ given $B$
• $P(A \cap B)$: Probability of both $A$ and $B$ occurring
• $P(B)$: Probability of $B$ occurring

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Solved Problems

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Problem 1

Question: In a class of 50 students, 30 study Mathematics, and 20 study both Mathematics and Physics. If a student studies Physics, what is the probability that they also study Mathematics?
Solution:

$$P(\text{Mathematics | Physics}) = \frac{P(\text{Mathematics} \cap \text{Physics})}{P(\text{Physics})}$$ $$P(\text{Physics}) = \frac{20}{50} = 0.4,\ P(\text{Mathematics} \cap \text{Physics}) = \frac{20}{50} = 0.4$$ $$P(\text{Mathematics | Physics}) = \frac{0.4}{0.4} = 1$$

Answer: $P(\text{Mathematics | Physics}) = 1$

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Problem 2

Question: A die is rolled. If the outcome is an even number, what is the probability that it is a 4?
Solution:

$$P(4 | \text{Even}) = \frac{P(4 \cap \text{Even})}{P(\text{Even})}$$

• $P(4 \cap \text{Even}) = P(4) = \frac{1}{6}$
• $P(\text{Even}) = \frac{3}{6} = \frac{1}{2}$

$$P(4 | \text{Even}) = \frac{\frac{1}{6}}{\frac{1}{2}} = \frac{1}{3}$$

Answer: $P(4 | \text{Even}) = \frac{1}{3}$

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Problem 3

Question: A box contains 5 red, 3 green, and 2 blue balls. If a ball is randomly picked and found to be red, what is the probability that it is one of the 3 red balls numbered 1, 2, or 3?
Solution:

$$P(\text{Numbered Red | Red}) = \frac{P(\text{Numbered Red} \cap \text{Red})}{P(\text{Red})}$$

• $P(\text{Red}) = \frac{5}{10} = 0.5$
• $P(\text{Numbered Red} \cap \text{Red}) = \frac{3}{10} = 0.3$

$$P(\text{Numbered Red | Red}) = \frac{0.3}{0.5} = 0.6$$

Answer: $P(\text{Numbered Red | Red}) = 0.6$

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Problem 4

Question: A student is selected randomly. The probability that they study both Chemistry and Biology is 0.2, and the probability that they study Biology is 0.5. What is the probability that a student studies Chemistry, given that they study Biology?
Solution:

$$P(\text{Chemistry | Biology}) = \frac{P(\text{Chemistry} \cap \text{Biology})}{P(\text{Biology})}$$ $$P(\text{Chemistry | Biology}) = \frac{0.2}{0.5} = 0.4$$

Answer: $P(\text{Chemistry | Biology}) = 0.4$

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Problem 5

Question: In a company, 60% of employees are male, and 70% of males are engineers. What is the probability that a randomly chosen engineer is male?
Solution: Let $M$ = Male, $E$ = Engineer.

$$P(M|E) = \frac{P(M \cap E)}{P(E)}$$

• $P(M) = 0.6, P(E|M) = 0.7$

$$P(M \cap E) = P(M) \cdot P(E|M) = 0.6 \cdot 0.7 = 0.42$$

Assume $P(E) = 0.5$ (given context or additional data).

$$P(M|E) = \frac{0.42}{0.5} = 0.84$$

Answer: $P(M|E) = 0.84$

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Summary

1. Conditional probability measures the dependency of events.
2. Formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$.
3. Practice problems enhance understanding of real-life applications.

UNIT I Bayes Theorem

 

Bayes' Theorem for Civil Engineering Students with a Rural Background