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- Statistics - Discussion
Statistics - Geometric Distribution
Geometric distribution is a special case of negative binomial distribution with 1 success out of specified,n trials. In other words, it is a probability distribution of number of occurences of successes and failures in a sequence of independent trails before a first success occurs. Following are the key points to be noted about a geometric experiment.
The experiment should be of x repeated trials.
Each trail have two possible outcome, one for success, another for failure.
Probability of success is same on every trial.
Output of one trial is independent of output of another trail.
Experiment should be carried out until first success is observed.
Geometric distribution probability can be computed using following:
Formula
${ g(x; P) = P \times (1-P)^{x-1} }$
Where −
${x}$ = Total number of trials.
${P}$ = Probability of success on each occurence.
${1-P}$ = Probability of failure on each occurence.
${g(x; P)}$ = Geometric distribution probability, the probability that an x-trial negative binomial experiment results in the first success on the xth trial, when the probability of success on each trial is P.
Example
Robert is a football player. His success rate of goal hitting is 70%. What is the probability that Robert hits his first goal on his fifth attempt?
Solution:
Here probability of success, P is 0.70. Number of trials, x is 5. Using geometric distribution formula, let's compute the probability of hitting first goal in fifth attempt.
${ g(x; P) = P \times (1-P)^{x-1} \\[7pt] \implies g(x; P) = 0.7 \times 0.3^4 \\[7pt] \, = 0.7 \times 0.0081 \\[7pt] \, = 0.00567 }$
Thus probability of hitting first goal in fifth attempt is $ { 0.00567 }$.
