Probability Distributions | Binomial
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Binomial Probability Calculator
| Cumulative? | Number of Successes | Number of Trials | Probability of Success on a Trial | ||||
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Binomial Theory
A binomial experiment has the following properties:
- The experiment consists of n repeated trials
- Each trial can result in just two possible outcomes. We call one of those outcomes a success and the other a failure.
- The probability of success, denoted by p, is the same on every trial.
- The trials are independent; that is, the outcome of one trial does not affect the outcome of other trials.
Thus, if we equate this to the flipping of a coin, where getting “heads” is a “success”, then the probabilities of both success and failure should be 1 / 2.
Now, if you flip a coin 100 times, what is the probability that you would get 43 heads? To answer that, there is a formula.
P(x) = n!⁄x!(n-x)! • px • (1-p)n-x
Here, n! means Factorial n. A Factorial is where you recursively multiply a number by decreasing values of itself. For instance, the Factorial of 5 is 5 x 4 x 3 x 2 x 1.
P means the probability.
n is the number of trials.
x is the number of successes we are interested in.
Calculating will provide you with the probability that success will occur x times over n tries. However, what if you don’t want to know the probability of receiving a specific number of successes, but rather a number of successes or less?
To do this, you simply use cumulative calculating, whereby you calculate for all the potential success outcomes and add them together. Thus, if you want to know the probability of getting 5 heads or less from 10 tries, then perform the calculation with P(5) + P(4) + P(3) + P(2) + P(1) + P(0). Note that calculating zero successes must also be carried out, since this is a possibility.
Binomial Distribution
Finding the Standard Deviation for Binomials is a little different. You will need to know how to do this for p and np charts.
Mean (µx)
n • p
Thus, the mean is probability times the number of trials.
Variance (σ2x)
n • p • (1 - p)
Thus, the variance is mean times one minus probability.
Standard Deviation (σx)
√ n • p • (1 - p)
Thus standard deviation is the square-root of the variance.