Binomial Distribution

Introduction

TBA

Density function

The density function for the Binomial distribution is $$f(x | K, p) = { K \choose x } p^x (1 - p) ^ { (K - x) }$$

Consider the density function for the Binomial distribution formed by summing $K =$ $\, 10 \,$ independent Bernoulli random variables, each with probability $p =$ $\, 0.5 \,$ of observing a $1$.

Specific to the plot above, the density function is $$f(x | K = 10, p = 0.5) = { 10 \choose x } 0.5^x (1 - 0.5) ^ { (K - x) }$$

Examples

1. Suppose an electronics manufacturing company produces microchips that are used in various electrical devices. Based on quality control tests, it's found that 95% of the microchips are free from defects, meaning they operate as expected. An electrical engineer randomly selects 10 microchips from the production line to use in a prototype device.
   1. What is the probability that all 10 microchips selected are free from defects?
   2. What is the probability that exactly 8 out of the 10 microchips selected are free from defects?
2. A city has 50 bridges that need to be inspected for safety. From past data, it is known that any given bridge has a 10% chance of failing the inspection due to significant wear and tear.
   1. What is the probability that exactly 5 bridges will fail the inspection?
   2. What is the probability that at least 3 bridges will fail the inspection?
3. At a certain school, 70% of students prefer chocolate ice cream over vanilla. If a random sample of 10 students is taken,
   1. what is the probability that exactly 7 students prefer chocolate?
   2. what is the probability that at least 8 students prefer chocolate?

Calculator

$X \sim$Binomial( $K =$ $\, 10 \,$, $p =$ $\, 0.5 \,$). $\mathbb{P}[X$ $\le$ $\, 5 \,$ $] = 0.62$.

References

Binomial distribution. Wikipedia: The Free Encyclopedia. Wikimedia Foundation, Inc. Accessed 02/10/2023.