### 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] 0.375`

**Example 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] 0.875`

Here is simpler code for cumulative probability:

**pbinom(x.orfewersuccesses, n.trials, p.probability)**

`pbinom(2,3,0.5)`

`## [1] 0.875`