Binomial Distributions

Binomial distributions meet the following 4 criteria, which can be memorised through the acronym BIDS:
1. Binary or in other words, only 2 possible outcomes, such as passing or failing a test or flipping a coin.
2. Independent trials, meaning that outcomes of different trials do not affect each other.
3. Defined number of trials.
4. Same probability for ALL TRIALS.
So, if you draw 3 cards, wanting to get the ace of heart, this is actually NOT an example of a binomial distribution because each card draw, changes the probability of the remaining card draws. Throwing a die three times, wanting to get sixes on all three throws on the other hand, IS an example of a binomial distribution because the probabilities remain fixed between trials.
For perhaps a more relevant example to you students out there, let’s imagine that you take a multiple choice test that has 4 answer alternatives with one correct answer for each question and 10 total questions. What is the probability that you barely get a passing grade. In other words, what are the chances that you answer exactly 5 questions correct while you just answer questions at random?
To calculate this we use the quite nasty looking binomial formula: P(X) = nCx p^x q^n-x, where:
- n = number of trials → 10
- x = number of successful events → 5
- p = probability of successful event → 0.25
- q = probability of UNsuccessful event → 0.75
But don’t worry it looks much worse than it is. Let’s start with this: nCx. This is called n Choose x and it’s used to gain the total amount of combinations in which we can answer EXACTLY 5 questions correctly. Since there is a total of 10 questions, we could answer the five first ones correctly, OR we could answer the five final questions correctly or anything in between. This part of the formula lets us count all of these possible alternatives.
So 10C5 is written out as: 10! / ((10-5)! x 5!). In case you are not familiar with exclamation marks in your equations, it is just a factorial and it is quite simple to use. So in this example, 10! is the same as 10 x 9 x 8 x 7… all the way to x 1. This means we can simplify this equation further:
(10 x 9 x 8 x 7 x 6 x 5!) / (5! x 5!) → (10 x 9 x 8 x 7 x 6 x ~~5!)~~ / (5! x ~~5!~~) → (10 x 9 x 8 x 7 x 6) / (5 x 4 x 3 x 2 x 1) = 252
Don’t worry this was the hardest part, now we can plug in this number along with the others into the formula to get 252 x 0.25^5 x 0.75^(10-5). So this part gives us the total amount of possible outcomes, this part gives us the probability of EXACTLY 5 successes AND 5 failures. So when we plug the number in we get 0.0584 or 5.84%
Now if you want to calculate the mean and standard deviation, the formula for those are:
- Mean = np → 10 x 0.25
- Standard deviation = square root of npq
When we plug the numbers from this example we find that the mean in this case is 2.5 and standard deviation is 1.369, which means that if a lot of people answer these test question randomly, on average, the amount of questions they answer correctly is 2.5 with a standard deviation of 1.369 questions.

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