probability

1. APPLICATION OF PROBABLITY
IN ENGINEERING
- BY GROUP 4

2. What is probability distribution?
• Probability distribution is a mathematical function that describes the likelihood of occurrence
of different outcomes in a random event. In other words, it is a way of representing the
uncertainty associated with a random variable. A random variable is a variable that can take
on different values based on the outcome of a random event.
• A probability distribution assigns a probability to each possible outcome of a random event. The
probabilities are usually represented as values between 0 and 1, where 0 represents an
impossible outcome and 1 represents a certain outcome. The sum of all probabilities in a
probability distribution must equal 1, since one of the possible outcomes must occur.
• There are different types of probability distributions, including discrete distributions and
continuous distributions. Discrete distributions, such as the Bernoulli and binomial
distributions, are used to model events with a finite number of outcomes. Continuous
distributions, such as the normal and exponential distributions, are used to model events with
an infinite number of possible outcomes.
• The shape of a probability distribution is determined by its parameters, which can be
estimated from data or derived from physical laws. Once a probability distribution has been
determined, it can be used to make predictions about the outcome of future events, estimate
the likelihood of specific outcomes, and perform various statistical analyses.
•

3. Types of probability distribution
1. Discrete Distributions: These distributions are used to model events with a finite or countably
infinite number of outcomes. Examples include the Bernoulli, Binomial, Poisson, and
Geometric distributions.
2. Continuous Distributions: These distributions are used to model events with an uncountably
infinite number of outcomes, such as real numbers. Examples include the Normal, Exponential,
Uniform, and Log-Normal distributions.
3. Univariate Distributions: These distributions describe the behavior of a single random
variable. Examples include the Normal and Exponential distributions.
4. Multivariate Distributions: These distributions describe the behavior of multiple random
variables. Examples include the Multivariate Normal and Dirichlet distributions.
5. Symmetric Distributions: These distributions have a mean that is equal to the median, and the
distribution is symmetrical around the mean. Examples include the Normal and Uniform
distributions.
6. Skewed Distributions: These distributions have a mean that is not equal to the median, and
the distribution is not symmetrical around the mean. Examples include the Log-Normal and
Pareto distributions.

4. NORMAL
DISTRIBUTION
- BONUS EXAMPLE USING A PYTHON
PROGRAM

5. BASICS OF NORMAL
DISTRIBUTION:
 A Normal Distribution is also known as a Gaussian distribution.
 The normal distribution is magical because most of the naturally
occurring phenomenon follows a normal distribution.
For example, blood pressure, IQ scores, heights follow the normal
distribution

6. STANDARD NORMAL
DISTRIBUTION AND Z-SCORE:
 The standard normal distribution has a mean of 0 and variance of 1.
 Any normal distribution can be converted to standard normal distribution to find
probability. In order to do this we use the ‘Z-SCORE’.
 EMPERICAL RULE: It states that 68% of the values of a normal distribution of
data lie within ‘1σ’ of the mean, 95% within ‘2σ’, and 99.7% within ‘3σ’.

7. PLOTTING A NORMAL CURVE
USING PYTHON:
 Suppose we have data of the heights of adults in a town which follows normal
distribution, we have a sufficient sample size with mean equal to 5.3 and the
standard deviation 1.
Here, loc = mean = 5.3
scale = standard deviation = 1

8. CALCULATING PROBABILITY:
A] X<4.5 B] 4.5≤ X ≤6.5 C] X>6.5
*CODE:

9. REAL WORLD APPLICATIONS:
 In the investment world, the periodic (daily, monthly, even annual) returns of
assets like stocks and bonds are assumed to follow a normal distribution.
 In the corporate world, the distribution of the severity of manufacturing defects
was found to be normally distributed (this makes sense: usually you make it
right, a few times you make it slightly wrong, and once in a blue moon you
completely mess it up)
 The process improvement framework Six Sigma was basically built around this
observation.
 In data science and statistics, statistical inference (and hypothesis testing)
relies heavily on the normal distribution.

12. Binomial Distribution
Q.What ia binomial distribution?
The binomial distribution is one of the most commonly used distributions in
statistics. It describes the probability of obtaining k successes in n binomial
experiments.
If a random variable X follows a binomial distribution, then the probability
that X = k successes can be found by the following formula:
P(X=k) = nCk * pk * (1-p)n-k
where:
•n: number of trials
•k: number of successes
•p: probability of success on a given trial
•nCk: the number of ways to obtain k successes in n trials

13. Generating an array that follows binomial distribution
using python
Each number in the resulting array represents the number of
“successes” experienced during 10 trials where the probability of
success in a given trial was 0.25.

14. Eg.Q1:Tanish makes 60% of his free-throw attempts. If he
shoots 12 free throws, what is the probability that he makes
exactly 10?
The probability that Tanish makes exactly 10 free throws
is 0.0639.

15. Eg.Q2: Afaz flips a fair coin 5 times. What is the probability
that the coin lands on heads 2 times or fewer?
The probability that the coin lands on heads 2 times or
fewer is 0.5.

16. Visualizing a Binomial Distribution
Code:
The x-axis describes the number of successes during 10 trials and the y-axis
displays the number of times each number of successes occurred during 1,000
experiments.