I am writing these quick notes for my didactic purposes and to provide useful starting places for my peers in grad school. If you see that I have made a mistake or would like to suggest some way to make the post better or more accurate, please feel free to email me. I am always happy to learn from others' experiences!

Table of contents

1. Bayes' Theorem
2. Going Through an Example

Bayes' Theorem

Bayes' Theorem is a fundamental concept in probability theory that allows us to update our beliefs about the probability of an event occurring given new information. The formula is written as:

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

where $P(A|B)$ is the probability of event A occurring given that event B has occurred, $P(B|A)$ is the probability of event B occurring given that event A has occurred, $P(A)$ is the prior probability of event A occurring, and $P(B)$ is the prior probability of event B occurring.

The formula can be interpreted as follows:

• $P(A|B)$ is the posterior probability of event A occurring given that event B has occurred.
• $P(B|A)$ is the likelihood of observing event B given that event A has occurred.
• $P(A)$ is the prior probability of event A occurring.
• $P(B)$ is the prior probability of event B occurring.

$P(B)$ can be calculated as:

$P(B) = P(B|A) * P(A) + P(B|\neg A) * P(\neg A)$

where $P(\neg A)$ is the probability of event A not occurring.

Going Through an Example

Let's go through an example to illustrate how Bayes' Theorem can be used to update our beliefs about the probability of an event occurring given new information.

Suppose you have two servers, Server 1 and Server 2. Server 1 hosts 70% of the website traffic and Server 2 hosts the remaining 30%. Server 1 has a 98% uptime rate, while Server 2 has 96% uptime rate. A user reports that the website is down. What is the probability that the website is down because of Server 2?

Let's define the events as follows:

• A: Server 1 hosts the website;
• B: Server 2 hosts the website;
• D: The website is down.

We want to calculate $P(A|D)$, the probability that Server 1 hosts the website given that the website is down.

We can calculate $P(A|R)$ using Bayes' Theorem as follows:

$P(A|D) = \frac{P(D|A) * P(A)}{P(D)}$

where $P(D|A)$ is the probability of the website being down given that Server 1 was hosting it, $P(A)$ is the prior probability of Server 1 hosting the site, and $P(D)$ is the prior probability of the site being down.

We can calculate $P(D|A)$ as the probability of the website being down given it's hosted by Server 1 $.02$, and $P(A)$ as the prior probability of the website being hosted by Server 1, which is $.7$.

To calculate $P(D)$, we can use the law of total probability:

$P(D) = P(D|A) * P(A) + P(D|B) * P(B)$

where $P(D|A)$ is the probability of the website being down given that it was hosted on Server 1, which is $.02$. $P(A)$ is the prior probability that the website was hosted on Server 1, which is $.7$. $P(D|B)$ is the probability of the website being down given that it was hosted on Server 2, which is $.04$, and $P(B)$ is the prior probability of the website being on Server 2, which is $.3$.

Substituting the values into the formula, we get:

$P(D) = .7 * .02 + .3 * .04 = .026$

Now we can substitute the values into the formula for $P(A|R)$:

$P(A|R) = \frac{.02 * .7}{.026} \approx 0.5384615385$

Therefore, the probability that the website is down because of Server 1 is approximately $.54$.