The formula for the Binomial Probability of exactly r success is nCr (p^r) (q^{n-r}) on page 216 of your textbook. The nCr in at the front is the number of combinations (without order) of r objects taken from n. Why is this a combination and not a permutation (counting with order)? In other words, why is it not nPr (p^r) (q^{n-r})?
The Correct Answer and Explanation is :
The formula for the Binomial Probability is given as:
[
P(X = r) = \binom{n}{r} p^r q^{n-r}
]
where:
- (\binom{n}{r}) is the number of combinations of (r) successes from (n) trials,
- (p) is the probability of success,
- (q = 1 – p) is the probability of failure,
- (r) is the number of successes, and
- (n – r) is the number of failures.
Now, the reason we use combinations ((\binom{n}{r})) and not permutations ((P(n, r))) in the Binomial Probability formula is rooted in the nature of the problem and the concept of “order.”
Combinations vs. Permutations:
- Combinations: When we are dealing with combinations, we are interested in how many ways we can select a group of objects without considering the order. For the binomial distribution, we are asking how many different ways (r) successes can occur in (n) trials, but the order in which the successes occur does not matter. For example, if we flip a coin 3 times and get 2 heads and 1 tail, the order of the flips doesn’t matter—only the number of heads and tails is important.
- Permutations: On the other hand, permutations are used when the order of the objects matters. If we were counting how many different ways (r) successes could occur where the order of success matters, then we would use permutations. But in the case of the Binomial distribution, we are not concerned with the specific order of the successes, but only with how many successes occur overall.
Why (\binom{n}{r}) in Binomial Distribution:
In the binomial experiment (such as flipping a coin (n) times), we are interested in the number of successes (heads in the case of a coin flip), regardless of the order in which they occur. For instance, the outcomes “HHH” and “HHT” both represent 2 successes, and thus they are treated as equivalent in a binomial experiment. This means we need to count the number of ways to choose (r) successes from (n) trials, which is exactly what a combination formula does. Hence, we use (\binom{n}{r}), which gives the number of ways to choose (r) successes (heads) from (n) trials.
If we used (P(n, r)), it would count different permutations (orders) of the successes, which is unnecessary in the binomial context because the order of the successes doesn’t matter—only the total number of successes and failures does.
Thus, the Binomial distribution is based on combinations, not permutations, because we care about how many successes occur, not the specific order in which they happen.