You guess all 20 questions on your multiple-choice exam. What are the chances that you pass?
Which student hasn’t been there? It’s exam time and the multiple-choice questions are waiting for you. Can’t be that hard to guess yourself to success, can it?
An exam has 20 questions. Each question has four choices. Exactly one choice is correct for each question. You need at least 10 correct answers to pass. What is the probability of passing if you pick one random answer for each question?
Before you read on (or scroll down to check the answer), I want you to make a rough guess. Just (mentally) tick one of the boxes above.
This is a common problem that can be described using the Binomial Distribution. And we encounter a lot of such problems in our everyday lives.
For example:
You roll the dice 3 times. What are the chances of getting exactly one six? What are the chances of getting at least one six?
You play the lottery every day for one year. What are the chances of winning at least once?
You make 3 penalty shots. What are the chances of hitting the goal at least two times?
The probability of having a certain disease is 1 %. What are the chances that in a group of 10 people at least two people have the disease?
All of these problems can be described using a Binomial Distribution. Before we look at it mathematically, let’s look at it from a more general perspective.
What do we need for a Binomial Distribution?
- one base experiment which will be repeated many times (n times).
- only two outcomes in the base experiment: success or no success; property fulfilled or not fulfilled.
- a success probability p that does not change from one experiment to the next (this means the experiments are independent).
What are we looking for?
We are looking for the probability of a certain number of successes. The order in which the successes happen does not matter. We don’t care whether we answered the first five questions or the last five questions on…
