Selected Discrete Probability Distributions
The Binomial Probability Distribution
\[f(x) = \binom{n}{x}p^{x}(1-p)^{n-x}\] n-number of independent trials, n = 0,1,2,…
x-number of successes in n trials, x = 0,1,2,…n
p-probability of success in single trial
In general, R-code for the binomial:
dbinom(x.successes, n.trials, p.probability)
Example 1
A fair coin was tossed three times and number of HEADS was observed. Determine the probability of “getting” two HEADS in three tosses.
n=3, x=2, p=0.5
\[f(2) = \binom{3}{2}0.5^{2}(1-0.5)^{3-2}\]
Here is corresponding R-code:
dbinom(2,3,0.5)## [1] 0.375Example 2
A fair coin was tossed three times and number of HEADS was observed. Determine the probability of “getting” two or fewer HEADS in three tosses.
n=3, x=0:2, p=0.5
\[f(0) + f(1) + f(2)\]
sum(dbinom(0:2,3,0.5))## [1] 0.875Here is simpler code for cumulative probability:
pbinom(x.orfewersuccesses, n.trials, p.probability)
pbinom(2,3,0.5)## [1] 0.875