The areas of statistics and probability are my favorite in mathematics. This is because I like to be able to draw conclusions from data and predict the future with past trends. Probability is also fascinating to me since it allows us to make more educated decisions about real-life events. Since we are supposed to get a big snow storm in Waterloo, I thought I would write a blog post discussing conditional probability using the Probability Tree Generator, created by Miles Simmons.

If the probability of snowfall on any given day during a Waterloo winter is 0.75, the probability that the schools are closed given that it has snowed is 0.6, and the probability that the schools are closed given that it has hasn’t snowed is 0.1, then we get the following probability tree, created by Miles’s learn document:

From this information we can come to some interesting conclusions:

What is the probability that the schools are closed on a given day?

From the Law of total probability, we get:

Thus, during a very snowy Waterloo winter, we could expect a 0.475 chance of schools being closed on any given day. 

One of the features of this document is that the node probabilities are calculated. You can see this by comparing the second last step to the number at the end of probability trees' nodes.

What is the probability that it has snowed given that the schools are closed?

From Bayes’ Theorem, we get:

Thus, during a very snowy Waterloo winter, we expect there to be a probability of 0.947 that it has snowed if the schools are closed. 

We can also add more events to the tree. For example, if the students are happy or sad given that the schools are open:

Even though we would all love schools to be closed 47.5% of the winter days in Waterloo, these numbers were just for fun. So, the next time you are hoping for a snow day, make sure to wear your pajamas inside out and sleep with a spoon under your pillow that night!

To explore more probability tree fun, be sure to check out Miles’s Probability Tree Generator, where you can create your own probability trees with automatically calculated node probabilities and export your tree to a blank Maple Learn document. Finally, if you are interested in seeing more of our probability collection, you can find it here!

In an age where our lives are increasingly integrated online, cybersecurity is more important than ever. Cybersecurity is the practice of protecting online information, systems, and networks from malicious parties. Whenever you access your email, check your online banking, or make a post on Facebook, you are relying on cybersecurity systems to keep your personal information safe. 

Requiring that users enter their password is a common security practice, but it is nowhere near hacker-proof. A common password-hacking strategy is the brute-force attack. This is when a hacker uses an automated program to guess random passwords until the right one is found. The dictionary attack is a similar hacking strategy, where guesses come from a list like the 10,000 Most Common Passwords.

The easiest way to prevent this kind of breach is to use strong passwords. First, to protect against dictionary attacks, never use a common password like “1234” or “password”. Second, to protect against brute-force attacks, consider how the length and characters used affect the guessability. Hackers often start by guessing short passwords using limited types of characters, so the longer and more special characters used, the better.

Using the Strong Password Exploration Maple Learn document, you can explore how susceptible your passwords may be to a brute-force attack. For example, a 6-character password using only lowercase letters and numbers could take as little as 2 seconds to hack.

Whereas an 8-character password using uppercase letters, lowercase letters, and 10 possible special characters could take more than 60 hours to crack.

These hacking times are only estimations, but they do provide insight into the relative strength of different passwords. To learn more about password possibilities, check out the Passwords Collection on Maple Learn