Quick Geometric Distribution Calculator

A geometric distribution can be defined as the probability of experiencing the number of failures before you get the first success in a series of Bernoulli trials. Bernoulli trials refer to two possible outcomes for each trial (success or failure). For example : What's the probability that we have to face 4 failures before we get heads on a coin.

$$P(X=k) = (1-p)^{k-1}p$$

where k is the number of trials. k-1 can be read as the number of failures prior to the first success. p is the probability of success on each trial.

Probability of Success (p) Number of trials (k)

Solution

$$P(X=9) = 0.001953$$ $$P(X \lt9) = 0.996094$$ $$P(X \le9) = 0.998047$$ $$P(X \gt9) = 0.001953$$ $$P(X \ge9) = 0.003906$$

P(X < 9) means probability of less than 9 failures before the first success. Similarly P(X > 9) refers to probability of more than 9 failures before the first success.

How to calculate P(X < 9)

To calculate P(X < 9), we need to sum all the probabilities from P(X = 0) through P(X = 9). See the details below.

$P(X \lt9)$ = P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6) + P(X=7) + P(X=8)

$P(X \lt9)$ = 0.500000 + 0.250000 + 0.125000 + 0.062500 + 0.031250 + 0.015625 + 0.007813 + 0.003906

$P(X \lt9) = 0.996094$

Geometric Distribution Table

$$P(X=1) = 0.500000$$ $$P(X=2) = 0.250000$$ $$P(X=3) = 0.125000$$ $$P(X=4) = 0.062500$$ $$P(X=5) = 0.031250$$ $$P(X=6) = 0.015625$$ $$P(X=7) = 0.007813$$ $$P(X=8) = 0.003906$$ $$P(X=9) = 0.001953$$

Important Points

• Probability of success must lie between 0 and 1.
• Number of trials can't be less than or equal to 0. It must be a whole number, can't be in decimals.

FAQs

Geometric Distribution has the following properties -

• Each trial has only two possible outcomes - success or failure.
• The probability of a success on any trial is same.
• Trials are independent.

Both have the same properties but they are different in terms of objective. In a binomial distribution, the number of trials is fixed. Whereas when we use a geometric distribution, we are interested in the number of trials required until we get success.

• Suppose you work as a operational manager in a factory and want to know the probability that the kth product on a production line is defective
• As a marketing lead, it is expected from you to calculate the number of trials required before sale turn out to be a success

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