The binomial distribution
I think the binomial distribution is the most fun of all distributions. In this section you will need your skills in probability theory. Here I elaborate on the example about Heads and Tails. The binomial distribution can for example be used to check if a coin really is fair; is the probability of a Head really 0.5?
In the binomial distribution you only need to be concerned about two outcomes; Head or Tail, yes or no, present or absent, 1 or 0. From every single toss of a coin, you can have the outcome of either a Head or a Tail. Every time you ask someone if he or she likes broccoli you can get either yes or no. This variable is therefore discrete; it is on the nominal scale. Head or Tail does not have an internal order.
The binomial distribution id defined by the following equation:
where is the probability of getting a combination of a specific outcome times in trials, is the probability of the outcome and =is the probability of the other outcome.
This equation looks quite intimidating at a first glance, but it is quite straightforward actually. What the equation does is to calculate the probability of getting a combination with a specific number of outcomes on a specific number of trials.
If you toss a coin 3 times, what is the probability of getting 2 Heads when the probability of getting a Head is 0.5?
Answer: The probability is 0.375 to get a combination with 2 Heads in 3 tosses when p =0.5.
How to do it in R
Below I present the binomial distribution for 10 trials and p = 0.5:
When p = 0.5 the distribution is symmetric, but see what happens when p=0.2:
Now, the distribution is asymmetrical. Thus, the distribution only conforms to a perfect normal bell-shaped distribution when p =0.5.
Important to remember:
In the binomial distribution you are dealing with two outcomes that you can observe from a trial.
You can use the binomial distribution to answer the following question:
“I know the probability getting a Head with my coin is 0.5, what is the probability of getting 2 Heads in 3 tosses?”
More in depth
How does the equation for the binomial distribution work?
Ok, now let’s go on to look at some probabilities.
The probability of getting a Head in every toss is p= 0.5 and the probability of getting a Tail is 1-p=q= 0.5.
When tossing the coin, say, three times you can get one of the eight following observations:
HHH = 3 Heads
TTT = 3 Tails = 0 Heads
HHT = 1 Tail = 2 Heads
HTH = 1 Tail = 2 Heads
THH = 1 Tail = 2 Heads
TTH = 2 Tails = 1 Head
THT = 2 Tails = 1 Head
HTT = 2 Tails = 1 Head
The probability of each of these combinations is:
Do you see a pattern? If you are not concerned with in which order the Head and Tail comes in the three tosses there are really only four unique combinations:
So the probability of getting 3, 0, 2 or 1 Head in three tosses is equal to the sum of the probability of all combinations containing 3, 0, 2 or 1 Head:
To simplify this table:
But how are we calculating the number of combinations with number of Heads. We have just dealt with 3 tosses now, but what if we are tossing the coin 10 times. How many combinations do we get for 3 Heads? The answer is 120. And for 100 tosses? The answer is 161 700. From this it is clear that we don’t have time to make tables even when dealing with a number of tosses as low as 10.
So to calculate the probability of a specific number of Heads (X) in k tosses we are multiplying the number of combinations © containing that number of Heads with the probability of each of these combinations (C x pC). Do you follow? We are using one of the principles from the probability theory here; we want the probability for a combination containing 2 Heads. There are three combinations with 2 heads, so we add them: 0.125 + 0.125 + 0.125 = 0.125 x 3 = 0.375. The probability for all possible combinations adds up to 1: 0.125 + 0.375 + 0.375 + 0.125 = 1.
Now we are on the way to an equation for calculating the probability of X number of Heads in k tosses. Simplified this equation is P (X, k) = C x pc
To calculate the number of combinations () with number of a specific outcome (e.g. Heads) on number of trials (tosses) you use the following equation:
Calculate the number of combinations you can get containing 6 Heads on 20 tosses.
= 20, = 6
Use the equation:
The answer is: You can get 38 760 combinations with 6 Heads on 20 tosses
Ok, so now we have an equation to calculate the number of combinations ©. But how do we calculate the probability of each combination? This is the most straightforward part. You take the probability of the outcome (e.g. p = 0.5) and raise it to the power of the number of times you will receive the outcome (e.g. 3 times): . But for the rest of the tosses( ), you will receive the other outcome ().
On tosses you use the following equation:
where is the probability of one combination with nr of heads (or 1:s) in tosses, is the probability of getting a head (or 1) and is the probability of getting a tail (or 0).
Calculate the probability for each combination containing 6 Heads in 20 tosses where the probability of getting a Head in each toss is p = 0.5 and a Tail is q = 1- p = 0.5
= 20, = 6
Use the equation:
Answer: The probability for each combination containing 6 Heads in 20 tosses is 9.53 x 10-7
Now, we are getting somewhere. Then we combine these to equations (C and Pc). That is the equation for the binomial distribution:
where is the probability of getting a combination of a specific outcome times in trials, is the probability of the outcome and = 1 — is the probability of the other outcome.
Calculate the probability of getting a combination with 6 Heads in 20 tosses where the probability of a Head is p=0.5.
= 20, = 6
Use the equation for the binomial distribution:
Answer: The probability of getting a combination with 6 Heads on 20 tosses is 0.037.