Concept

Geometric Distribution

A geometric distribution is the probability distribution that describes the number of binomial trials needed in an experiment until the first success occurs. The trials must satisfy three conditions.

There are two possible outcomes for each trial — success and failure.
The trials are independent.
The probability of success is the same on each trial.

If $p$ is the probability of success and $n$ is the number of trials needed to get a success, the probability of getting a success in $n$ trials is given by the formula below.

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

Consider a situation in which a regular coin is flipped until it lands on heads.

It is worth noting that this experiment satisfies the three conditions because the probability of landing on heads is

0.5

for every trial. The following expression gives the probability of landing on heads after

n

trials.

P (X = n) P (X = n) = 0 . 5^{n - 1} 0.5 ⇓ = 0 . 5^{n}

The graph below displays the probabilities for values of

n

from

1

10 .

The value of

n = 1

indicates that the coin is flipped once before it lands on heads. The value of

n = 2

indicates that the coin lands on heads the second time it is flipped, and so on.

It should be noted that the probability of landing on heads after $n$ coin flips is equal to the probability of landing on tails after $n - 1$ consecutive coin flips, then landing on heads on the next flip.

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