The probability of success (symbolized p) is at the heart of the binomial probability distribution. This number is obtained on the basis of prior knowledge or theory. In the present example pertaining to pretrial release, we know that 62% of felony defendants obtain pretrial release. This means that each defendant’s probability of release is p = .62.
We know that release is not the only possible outcome, though—defendants can be detained, too. The opposite of a success is a failure. Because we are dealing with events that have two potential outcomes, we need to know the probability of failure in addition to that of success. The probability of failure is represented by the letter q ; to compute the value of q , the bound rule and the rule of the complement must be invoked. In Chapter 5, you learned that
p (A) + p (Not A) = 1.00; therefore,
p (Not A) = 1.00 – p (A) .
Failure: Any outcome other than success or the event of interest.
What we are doing now is exactly the same thing, with the small change that p (A) is being changed to simply p , and p (Not A) will now be represented by the letter q. So,
With q = .62, the probability that a felony defendant will be detained prior to trial (i.e., will not obtain pretrial release) is
q = 1.00 – .62 = .38
Putting It All Together: Using the Binomial Coefficient to Construct the Binomial Probability Distribution
Using p and q , the probability of various combinations of successes and failures can be computed. When there are r successes over N trials, then there are N – r failures over that same set of trials. There is a formula called the restricted multiplication rule for independent events that guarantees that the probability of r successes and N – r failures is the product of p and q. In the present example, there are three successes (each with probability p = .62) and two failures (each with probability q = .38). You might also recall from prior math classes that exponents can be used as shorthand for multiplication when a particular number is multiplied by itself many times. Finally, we also have to account for the combination of N and r. The probability of three successes is thus
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