Binomial experiment: Four Steps
Determining if a question concerns a binomial experiment involves asking yourself four questions about the problem. It’s as simple as that!
Sample question: which of the following are binomial experiments?
- Telephone surveying a group of 200 people to ask if they voted for George Bush.
- Counting the average number of dogs seen at a veterinarian’s office daily.
- You take a survey of 50 traffic lights in a certain city, at 3 p.m., recording whether the light was red, green, or yellow at that time.
- You are at a fair, playing “pop the balloon” with 6 darts. There are 20 balloons. 10 of the balloons have a ticket inside that say “win,” and 10 have a ticket that says “lose.”
Step 1:Ask yourself: is there a fixed number of trials?
- For question #1, the answer is yes (200).
- For question #2, the answer is no, so we’re going to discard #2 as a binomial experiment.
- For question #3, the answer is yes, there’s a fixed number of trials (the 50 traffic lights).
- For question #4, the answer is yes (your 6 darts).
Step 2: Ask yourself: Are there only 2 possible outcomes?
- For question #1, the only two possible outcomes are that they did, or they didn’t vote for Mr. Bush, so the answer is yes.
- For question #3, there are 3 possibilities: red, green, and yellow, so it’s not a binomial experiment.
- For question #4, the only possible outcomes are WIN or LOSE, so the answer is yes.
Step 3: Ask yourself: are the outcomes independent of each other? In other words, does the outcome of one trial (or one toss, or one question) affect another trial?
- For question #1, the answer is yes: one person saying they did or didn’t vote for Mr. Bush isn’t going to affect the next person’s response.
- For question #4, each time you toss a dart, the number of winning and losing tickets changes, which means, for example, if you win one toss, the probability of winning isn’t 10 to 10 anymore, but 9 to 10, since you already have one of the winning tickets. Since the probability is different, the trials are not independent, so the answer is no, and question #4 is not a binomial experiment.
Step 4: Does the probability of success remain the same for each trial?
- For question #1, the answer is yes, each person has a 50% chance of having voted for Mr. Bush.
Question #1 out of the 4 given questions was the only one that was a binomial experiment.
For more on coin toss examples, see this article at the University of Western Michigan.