
In a binomial experiment there are two mutually exclusive outcomes, often referred to as "success" and "failure". If the probability of success is p, the probability of failure is 1  p.
Such an experiment whose outcome is random and can be either of two possibilities, "success" or "failure", is called a Bernoulli trial, after Swiss mathematician Jacob Bernoulli (1654  1705). 

Examples of Bernoulli trials: 

• flipping a coin  heads is success, tails is failure


• rolling a die  3 is success, anything else is failure 

• voting  votes for candidate A is success, anything else is failure 

• determining eye color  green eyes is success, anything else is failure 

• spraying crops  the insects are killed is success, anything else is failure 
When computing a binomial probability, it is necessary to calculate and multiply three separate factors:
1. the number of ways to select exactly r successes,
2. the probability of success (p) raised to the r power, 3. the probability of failure (q) raised to the (n  r) power. 
The probability of an event, p, occurring exactly r times: 
n = number of trials r = number of specific events you wish to obtain p = probability that the event will occur q = probability that the event will not occur
(q = 1  p, the complement of the event) 

If we use an alternate notation for combination,
and express the complement value q as (1  p),
we have an alternate formula for
binomial probability. 

Alternative formula form 

The graphing of all possible binomial probabilities related to an event creates a binomial distribution. Consider the following distributions of tossing a fair coin:
Two Toss 
Four Toss

In the following examples, answers will be rounded to 3 decimal places.
When rolling a die 100 times, what is the probability of rolling a "4" exactly 25 times? 

Solution:
n = 100
r = 25
n – r = 75
p = 1/6 = probability of rolling a "4"
q = 1  p = 5/6 = probability of not rolling a "4"

At a certain intersection, the light for eastbound traffic is red for 15 seconds, yellow for 5 seconds, and green for 30 seconds. Find the probability that out of the next eight eastbound cars that arrive randomly at the light, exactly three will be stopped by a red light. 

Solution:
n = 8
r = 3
n – r = 5
p = 15/50 = probability of a red light
q = 1  p = 35/50 = probability of not a red light


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