# How to compute probabilities from a distribution (in R)

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There are many famous continuous probability distributions, such as the normal and exponential distributions. How can we get access to them in software, to compute the probability of a value/values occurring?

## Solution

Because R is designed for use in statistics, it comes with many probability distributions built in. A list of them is online here.

To compute a probability from a discrete distribution, prefix the name of the distribution with d (for “density”) and call it as a function on the value whose probability you want to know, plus any parameters the distrubtion needs.

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# For a binomial random variable with 10 trials
# and probability 0.5 of success on each trial,
# what is the probability of exactly 3 successes?
dbinom( 3, size=10, prob=0.5 )

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[1] 0.1171875


If you change the prefix to p, then R will compute the probability up to the parameter you specify, as in the following example.

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# For a binomial random variable with 10 trials
# and probability 0.5 of success on each trial,
# what is the probability of up to (and including) 3 successes?
pbinom( 3, size=10, prob=0.5 )

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[1] 0.171875


To compute a probability from a continuous distribution, prefix the name with d, just as in the example above. But you can compute only the probability that a random value will fall in an interval $[a,b]$, not the probability that it will equal a specific value.

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# For a normal random variable with mean μ=10 and standard deviation σ=5,
# what is the probability of the value lying in the interval [12,13]?
pnorm( 13, mean=10, sd=5 ) - pnorm( 12, mean=10, sd=5 )

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[1] 0.07032514


Consequently, we can also compute:

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pnorm( 13, mean=10, sd=5 )     # the probability of a value < 13
1 - pnorm( 13, mean=10, sd=5 ) # the probability of a value > 13

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[1] 0.7257469

[1] 0.2742531


See a problem? Tell us or edit the source.

Contributed by:

• Nathan Carter (ncarter@bentley.edu)
• Elizabeth Czarniak (CZARNIA_ELIZ@bentley.edu)