# How to Calculate Geometric Probabilities

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}^{ }^{– 1}*p* 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*)^{x}^{ – }^{1}*p*. 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*)^{x}*p*

*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*)^{x}*p*. 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.