## 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.

**Example **

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*

dbinom(2,3,prob=0.5)

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 p_{C}).** 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 *p*_{c}

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:

**Example**

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).

**Example**

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 *P _{c}*). 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.

**Example **

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.