 The geometric distribution is based on the binomial process (a series of independent trials with two possible outcomes). You use the geometric distribution to determine the probability that a specified number of trials will take place before the first success occurs. Alternatively, you can use the geometric distribution to figure the probability that a specified number of failures will occur before the first success takes place.

To calculate the probability that a given number of trials take place until the first success occurs, use the following formula:

P(X = x) = (1 – p)x – 1p for x = 1, 2, 3, . . .

Here, x can be any whole number (integer); there is no maximum value for x.

X is a geometric random variable, x is the number of trials required until the first success occurs, and p is the probability of success on a single trial.

For example, suppose you want to flip a coin until the first heads turns up. The probability that it takes four flips for the first heads to occur (that is, three tails followed by one heads) is P(X = x) = (1 – p)x1p. In this example, x = 4 and p = 0.5:

P(X = 4) = (1 – 0.5)3(0.5) = (0.125)(0.5) = 0.0625

To calculate the probability that a given number of failures occur before the first success, the formula is

P(X = x) = (1 – p)xp

x now represents the number of failures that occur before the first success. In addition, x can assume values 0, 1, 2, . . . instead of 1, 2, 3, . . .

For example, suppose you flip a coin until the first heads turns up. The probability that there will be three tails before the first heads turns up is P(X = x) = (1 – p)xp. In this example, x = 3 and p = 0.5:

P(X = 3) = (1 – 0.5)3(0.5) = (0.5)3(0.5) = (0.125)(0.5) = 0.0625

Both situations refer to getting three tails followed by a heads, so both formulas provide the same result.