1. The binomial distribution
Now let's learn about another distribution - the binomial distribution.
2. Coin flipping
We'll start by flipping a coin, which has two possible outcomes, heads or tails, each with a probability of fifty percent.
3. Binary outcomes
This is an example of a binary outcome, where two possible values can occur. We could also represent these outcomes as a one or a zero, a success or a failure, and a win or a loss.
4. One coin flip many times
We can perform multiple flips of a single coin and log the results, such as here where heads is represented as a one, and tails as a zero.
5. Binomial distribution
The binomial distribution describes the probability of the number of successes in a sequence of independent events. For example, it can tell us the probability of getting some number of heads in a sequence of coin flips. Note that this is a discrete distribution since we're working with a countable outcome.
The binomial distribution can be described using two parameters, n and p. n represents the total number of events being performed, and p is the probability of success, in this case, heads.
6. Binomial distribution
Here's what the distribution looks like for 10 coin flips. We have the biggest chance of getting five heads, and a much smaller chance of getting zero or 10 heads.
7. Probability of 7 or fewer heads
As with other distributions, we can calculate the probability of outcomes by adding together the area.
If we want the probability of getting seven or fewer heads from 10 flips, we add up the probability of rolling zero heads, one head, two heads and so on up to and including seven heads.
There is a 94.5 percent probability of rolling seven or fewer heads.
8. Probability of 8 or more heads
Likewise, to calculate the probability of eight or more heads, we can subtract the probability of seven or fewer heads from the total probability, or one.
This gives a result of around 5.5 percent probability.
9. Expected value
The expected value of the binomial distribution can be calculated by multiplying n by p. The expected number of heads from flipping 10 coins is 10 times 0.5, which is five.
If we don't know p, but know n and the expected value, we can calculate p through dividing the expected value by n.
10. Independence
In order for the binomial distribution to apply, each event must be independent, so the outcome of one event shouldn't have an effect on the next.
For example, if we're picking randomly from these cards with zeros and ones, we have a 50-50 chance of getting a zero or a one.
If probabilities change based on the outcome of a prior event, the binomial distribution does not apply.
11. Independence
But if we're sampling without replacement, the probabilities for the second event are different due to the outcome of the first event. Since these events aren't independent, we can't calculate accurate probabilities for this situation using the binomial distribution.
12. General applications
The binomial distribution can be used in any scenario where independent events produce binary outcomes, and does not require equal probability for each outcome.
Examples include clinical trials measuring effectiveness of a drug, where the outcome is whether the drug worked or not, or betting on the result of a sports match, where the bettor can either win or lose.
13. Let's practice!
Time to explore binary outcomes using the binomial distribution.