A binomial distribution is a probability distribution function used when there are exactly two mutually exclusive possible outcomes of a trial. The outcomes are classified as success and failure, and the binomial distribution is used to obtain the probability of observing x successes in n trials.
Intuitively, it follows that in the case of a binomial distribution, the random variable X can be defined to be the number of successes obtained in the trials.
Modelling binomial distributions
You can model X with a binomial distribution, B (n, p), if:
There are a fixed number of trials, n.
There are two possible outcomes, success and failure.
There is a fixed probability of success, p, for all trials.
The trials are independent.
If a random variable, X, has the binomial distribution B (n, p), then its probability mass function is given by:
\(P(X = r) = ^n{C}_r p^r(1-p)^{n-r}\).
Here are a few things to note about the above formula:
Let us look at an example of a binomial distribution to make things clearer.
Suppose the probability that a person selected at random likes butterscotch ice cream is 0.3. We select 100 people at random and ask each person if he/she likes butterscotch ice cream.
Solutions
Let us draw the binomial distribution for the above scenario.
Here,
p = 0.3
n=100
The following graph shows the binomial distribution for X ~ B (100, 0.3):
Graph for binomial distribution X ~ B (100, 0.3), Nilabhro Datta, StudySmarter Originals
Let's analyse the above binomial distribution in a bit more depth.
For any r (0 ≤ r ≤ 100), P (X = r) gives the probability that exactly r people would like butterscotch ice cream out of 100 randomly selected people.
From the given distribution, we can see that the most likely outcome is r = 30. The probability of 30 randomly selected people out of 100 liking butterscotch ice cream is 0.087.
The next most likely outcomes are r = 29 with a probability of 0.086 and r = 31 with a probability of 0.084. The probability of exactly 16 out of 100 people liking butterscotch ice cream is 0.00056. The probability of exactly 40 out of 100 people liking butterscotch ice cream is 0.0085. (The exact values are, of course, not decipherable from the above graph, but these are the values that were used to plot the graph).
A fair dice is tossed five times. Build a binomial distribution to find the probability of getting a certain number of heads.
Solution
Here, we can define a success as the event of obtaining a head in the trial. The probability of success = p = 0.5.
We have to build the binomial distribution function for X ~ B (5, 0.5).
For this example, let us build the binomial distribution by calculating the value of P (X = r) for each r by applying the formula for the probability mass function.
\(P(X = 0) = ^n{C}_rp^r(1-p)^{n-r} = ^5{C}_0 \cdot 0.5^0(1 - 0.5)^{5-0} = \frac{5!}{0!5!} \cdot 0.5^0 (0.5)^5 = 0.03125\)
\(P(X = 1) = ^n{C}_rp^r(1-p)^{n-r} = ^5{C}_1 \cdot 0.5^1(1 - 0.5)^{5-1} = \frac{5!}{1!4!} \cdot 0.5^1 (0.5)^4 = 0.15625\)
\(P(X = 2) = ^n{C}_rp^r(1-p)^{n-r} = ^5{C}_2 \cdot 0.5^2(1 - 0.5)^{5-2} = \frac{5!}{2!3!} \cdot 0.5^2 (0.5)^3 = 0.3125\)
\(P(X = 3) = ^n{C}_rp^r(1-p)^{n-r} = ^5{C}_3 \cdot 0.5^3(1 - 0.5)^{5-3} = \frac{5!}{3!2!} \cdot 0.5^3 (0.5)^2 = 0.3125\)
\(P(X = 4) = ^n{C}_rp^r(1-p)^{n-r} = ^5{C}_4 \cdot 0.5^4(1 - 0.5)^{5-4} = \frac{5!}{4!1!} \cdot 0.5^4 (0.5)^1 = 0.15625\)
\(P(X = 5) = ^n{C}_rp^r(1-p)^{n-r} = ^5{C}_5 \cdot 0.5^5(1 - 0.5)^{5-5} = \frac{5!}{5!0!} \cdot 0.5^5 (0.5)^0 = 0.03125\)
Therefore, the probability mass function will be
\(P(X=0) = P(X=5) = 0.03125; \quad P (X=1) = P(X=4) = 0.15625; \quad P(X=2) = P(X=3) = 0.3125\)
For the random variable X ~ B (8, 0.4), find
P(X=3)
2. P(X=0)
Solutions
1. \(P(X= 3) = ^n{C}_r p^r(1-p)^{n-r} = ^8{C}_3 \cdot 0.4^3(1- 0.4)^{8-3}= \frac{8!}{3! 5!} \cdot 0.4^3(0.6)^5 = 0.279\)
2. \(P (X=0) = ^n{C}_r p^r(1-p)^{n-r} = ^8{C}_0 0.4^0(1-0.4)^{8-0} = \frac{8!}{0!8!} \cdot 0.4^0(0.6)^8 = 0.017\)
Cumulative binomial distribution function
A cumulative probability distribution function for a binomial distribution, X ~ B (n, p) gives you the sum of all the individual probabilities up to and including the point, x, for the calculation for P (X ≤ x).
This implies that the cumulative probability at a point r of a binomial distribution will give the probability that the number of successes is less than or equal to r.
The formula for the binomial cumulative probability function is:
\(F(x, p, n) = \sum^x_{i=0}C_i p^{i}(1-p)^{n-i}\)
Let us revisit an earlier example, but this time we will draw the cumulative binomial distribution function.
Suppose the probability that a person selected at random likes butterscotch ice cream is 0.3. We select 100 people at random and ask each person if he/she likes butterscotch ice cream.
Solutions
Let us draw the cumulative binomial distribution function for the above scenario.
Here,
p = 0.3,
n=100.
The following graph shows the cumulative binomial distribution for X ~ B (100, 0.3):
Cumulative binomial distribution for X ~ B (100, 0.3), Nilabhro Datta, StudySmarter Originals
Let's analyse the above cumulative binomial distribution in a bit more depth.
For any r (0 ≤ r ≤ 100), P (X = r) gives the probability that r or fewer people would like butterscotch ice cream out of 100 randomly selected people.
The probability that 20 or fewer people would like butterscotch ice cream out of 100 is 1.65%. The probability that 30 or fewer people would like butterscotch ice cream out of 100 is 54.91%. The probability that 40 or fewer people would like butterscotch ice cream out of 100 is 98.75%. (The exact values are, of course, tough to decipher from the above graph, but these are the values that were used to plot the graph)
A fair dice is tossed five times. Create a table showing the cumulative binomial distribution to find the probability of getting less than or equal to a certain number of heads.
Solution 2
Here, we can define a success as the event of obtaining a head in the trial. The probability of success = p = 0.5.
No. of heads, r | 0 | 1 | 2 | 3 | 4 | 5 |
\( P(X = r) = ^n{C}_rp^r(1-p)^{n-r}\) | 0.03125 | 0.15625 | 0.3125 | 0.3125 | 0.15625 | 0.03125 |
Cumulative probability P(X ≤ r) | 0.03125 | 0.1875 | 0.5 | 0.8125 | 0.96875 | 1 |
Binomial Distribution - Key takeaways
A binomial distribution is used when there are exactly two mutually exclusive possible outcomes of a trial. The outcomes are classified as success and failure, and the binomial distribution is used to obtain the probability of observing x successes in n trials.
If a random variable, X, has the binomial distribution B (n, p), then its probability mass function is given by:
\(P(X=r)= ^n{C}_rp^r(1-p)^{n-r}\)
A cumulative probability distribution function for a binomial distribution, X ~ B (n, p) gives you the sum of all the individual probabilities up to and including the point x for the calculation for P (X ≤ x).
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