Jacob Bernoulli, also known as James or Jacques, was one of the prominent mathematicians from the Bernoulli family. He was the first pioneer in Leibnizian analysis and supported Leibniz in the calculus debate against Newton.
Jacob Bernoulli is famous for his many contributions to calculus. He was one of the founders of calculus of variations and proposed the first version of the law of large numbers in his book "Ars Conjectandi" published in 1713. One of the important discussions in that book was about binomial experiments.
Binomial distribution is actually a fairly simple concept if we understand it from the basics. Imagine you're conducting an experiment that only has two possible outcomes: success or failure. For example, like flipping a coin that can only result in heads or tails.
Binomial distribution is used when we perform the same experiment repeatedly under the same constant conditions. Importantly, each experiment is independent (they don't influence each other) - meaning the result of previous experiments doesn't affect subsequent experiments.
Requirements for binomial experiments:
There are only two possible outcomes: success or failure
The number of experiments is predetermined and fixed
Each experiment is mutually independent
The probability of success is the same for each experiment
If there's a binomial experiment with probability of success p and probability of failure q=1−p, then the formula to calculate the probability of getting exactly x successes in n independent experiments is:
b(x;n,p)=(xn)pxqn−x forx=0,1,2,...,n
Where:
n = number of experiments performed
x = number of successes we want
p = probability of success in one experiment
q=1−p = probability of failure in one experiment
So, to use this formula, make sure first that the experiment we're dealing with meets the binomial requirements mentioned earlier.
Let's look at a simple example. Suppose we have a fair coin. We call heads as "H" and tails as "T", so the sample space is S={H,T}.
If we consider getting tails as "success", then the probability of success is p=21. Automatically, the probability of failure (getting heads) is q=1−21=21.
Problem: If we flip this coin 7 times in a row, what's the probability of getting tails exactly 5 times?
Solution:
First, let's identify the parameters:
n=7 (number of flips)
x=5 (number of tails desired)
p=21 (probability of getting tails)
q=21 (probability of getting heads)
Now we use the binomial distribution formula:
b(5;7,21)=(57)(21)5(21)7−5
=(57)(21)5(21)2
=(57)(21)7
Let's calculate (57) first:
(57)=5!(7−5)!7!=5!⋅2!7!
=5!×2×17×6×5!=27×6=242=21
So the complete calculation is:
b(5;7,21)=21×271=21×1281=12821
Therefore, the probability of getting 5 tails in 7 flips is 12821 or approximately 16.4%.
In a bag there are 10 balls that are exactly the same, except for their color: 6 red balls and 4 blue balls. If we draw 5 balls one by one with replacement (each drawn ball is returned), what's the probability of drawing exactly 3 red balls?
An archer has an accuracy rate of 80% for hitting targets. If he shoots 6 times, what's the probability he hits the target exactly 4 times?
There's a multiple choice exam with 4 answer choices for each question. A student guesses all answers randomly for 8 questions. What's the probability he answers exactly 2 questions correctly?
