A binomial distribution is a technique in statistics that you use when:

• The trial in question only has 2 outcomes (yes / no)
• The probability of success for each trial is the same across attempts.
• You want to find the probability of successes versus failures in X attempts.

They can be used to evaluate the odds of something happening when you have the exact odds. Some would think that if something had a 10% chance to happen, then it would be guarenteed after 10 attempts, but this is not the case!

Example: Let’s say that a car has a 1% chance to break down when starting the ignition. It sounds like a pretty low chance, right? However, after only 100 attempts of starting the engine, its around a 64% chance that your car has broken down at least once! The probability of the car breaking down k times is given with the formula: \(P(X = k) = \binom{100}{k} (0.01)^k (0.99)^{100-k}\)

We’ll view this on the next slide as a graph using the Binomial PMF above, which shows the probability of each number of successes in a group of 100 trials.

On this graph, we can see that the probability of the car never breaking down is about 36%. 1% is quite a high probability in this context!

On this trial, I’ve lowered the odds to about 0.01%. As you can see above, the odds are much more in your favor!

Another example where this math comes into play is video games! A quite infamous game where drop rates are concerned is Runescape, a game focused around killing monsters for their valued drops. Some of these drops are quite common, ranging from 1/3 to 1/100. However, there are some extremely lucrative items with droprates in the thousands! One of the more lucrative drops, the Draconic Visage, is a staggering 1/10000 chance for it to drop.

The probability of getting at least one drop is given with this formula: \(P(X \ge 1) = 1 - P(X = 0) = 1 - (1-p)^n\)

On the next slide is a graph showing the chance for you to get at least one drop of this item, over 10000 monster kills. You’ll notice that even after 10000 kills, its not guarenteed that it drops for you!

If you want to play around with these yourself you can with RStudio and Plotly. Take the code used to create the previous plot as an example, plug in your own numbers and experiment away! As these slides are quite small, Ive put the code on the next slide.

p <- 1/10000   
n_max <- 10000 
attempts <- 1:n_max
prob_at_least_one <- 1 - (1 - p)^attempts
df <- data.frame(attempts = attempts, prob = prob_at_least_one)
plot_ly(
  df,
  x = ~attempts,
  y = ~prob,
  type = "scatter",
  mode = "lines",
  line = list(color = "firebrick", width = 2)
) %>%
  layout(
    title = "Probability of Getting At Least 1 Drop Over 10000 Attempts",
    xaxis = list(title = "Number of Attempts"),
    yaxis = list(title = "P", range = c(0, 1))
  )

Thank you for viewing my presentation. I hope you took something away from this, and that you can apply it to other real world scenarios.