Probability calculator

Calculate probabilities or values based on the *Binomial*,
*Chi-squared*, *Discrete*, *F*,
*Exponential*, *Normal*, *Poisson*, *t*, or
*Uniform* distribution.

Suppose Consumer Reports (CR) wants to test manufacturer claims about battery life. The manufacturer claims that more than 90% of their batteries will power a laptop for at least 12 hours of continuous use. CR sets up 20 identical laptops with the manufacturerâ€™s batteries. If the manufacturerâ€™s claims are accurate, what is the probability that 15 or more laptops are still running after 12 hours?

The description of the problem suggests we should select
`Binomial`

from the `Distribution`

dropdown. To
find the probability, select `Values`

as the
`Input type`

and enter `15`

as the
`Upper bound`

. In the output below we can see that the
probability is 0.989. The probability that exactly 15 laptops are still
running after 12 hours is 0.032.

A manufacturer wants to determine the appropriate inventory level for headphones required to achieve a 95% service level. Demand for the headphones obeys a normal distribution with a mean of 3000 and a standard deviation of 800.

To find the required number of headphones to hold in inventory choose
`Normal`

from the `Distribution`

dropdown and then
select `Probability`

as the `Input type`

. Enter
`.95`

as the `Upper bound`

. In the output below we
see the number of units to stock is 4316.

A **discrete** random variable can take on a limited
(finite) number of possible values. The **probability
distribution** of a discrete random variable lists these values
and their probabilities. For example, probability distribution of the
number of cups of ice cream a customer buys could be described as
follows:

- 40% of customers buy 1 cup;
- 30% of customers buy 2 cups;
- 20% of customers buy 3 cups;
- 10% of customers buy 4 cups.

We can use the probability distribution of a random variable to
calculate its **mean** (or **expected value**)
as follows;

\[ E(C) = \mu_C = 1 \times 0.40 + 2 \times 0.30 + 3 \times 0.20 + 4 \times 0.10 = 2\,, \]

where \(\mu_C\) is the mean number
of cups purchased. We can *expect* a randomly selected customer
to buy 2 cups. The variance is calculated as follow:

\[ Var(C) = (1 - 2)^2 \times 0.4 + (2 - 2)^2 \times 0.3 + (3 - 2)^2 \times 0.2 + (4 - 2)^2 \times 0.1 = 1\,. \]

To get the mean and standard deviation of the discrete probability distribution above, as well as the probability a customer will buy 2 or more cups (0.6), specify the following in the probability calculator.

You can also use the probability calculator to determine a
`p.value`

or a `critical value`

for a statistical
test. See the help files for `Single mean`

,
`Single proportion`

, `Compare means`

,
`Compare proportions`

, `Cross-tabs`

in the
*Basics* menu and `Linear regression (OLS)`

in the
*Model* menu for details.

Add code to
*Report
> Rmd* to (re)create the analysis by clicking the
icon on the bottom
left of your screen or by pressing `ALT-enter`

on your
keyboard.

If a plot was created it can be customized using `ggplot2`

commands (e.g.,
`plot(result) + labs(title = "Normal distribution")`

). See
*Data
> Visualize* for details.

For an overview of related R-functions used by Radiant for
probability calculations see
*Basics
> Probability*

Key functions from the `stats`

package used in the
probability calculator:

`prob_norm`

uses`pnorm`

,`qnorm`

, and`dnorm`

`prob_lnorm`

uses`plnorm`

,`qlnorm`

, and`dlnorm`

`prob_tdist`

uses`pt`

,`qt`

, and`dt`

`prob_fdist`

uses`pf`

,`qf`

, and`df`

`prob_chisq`

uses`pchisq`

,`qchisq`

, and`dchisq`

`prob_unif`

uses`punif`

,`qunif`

, and`dunif`

`prob_binom`

uses`pbinom`

,`qbinom`

, and`dbinom`

`prob_expo`

uses`pexp`

,`qexp`

, and`dexp`

`prob_pois`

uses`ppios`

,`qpois`

, and`dpois`

Copy-and-paste the full command below into the RStudio console (i.e., the bottom-left window) and press return to gain access to all materials used in the probability calculator module of the Radiant Tutorial Series:

usethis::use_course("https://www.dropbox.com/sh/zw1yuiw8hvs47uc/AABPo1BncYv_i2eZfHQ7dgwCa?dl=1")

Describing the Distribution of a Discrete Random Variable (#1)

- This video shows how to summarize information about a discrete random variable using the probability calculator in Radiant
- Topics List:
- Calculate the mean and variance for a discrete random variable by hand
- Calculate the mean, variance, and select probabilities for a discrete random variable in Radiant

Describing Normal and Binomial Distributions in Radiant(#2)

- This video shows how to summarize information about Normal and Binomial distributions using the probability calculator in Radiant
- Topics List:
- Calculate probabilities of a random variable following a Normal distribution in Radiant
- Calculate probabilities of a random variable following a Binomial distribution by hand
- Calculate probabilities of a random variable following a Binomial distribution in Radiant

Describing Uniform and Binomial Distributions in Radiant(#3)

- This video shows how to summarize information about Uniform and Binomial distributions using the probability calculator in Radiant
- Topics List:
- Calculate probabilities of a random variable following a Uniform distribution in Radiant
- Calculate probabilities of a random variable following a Binomial distribution in Radiant

Providing Probability Bounds(#4)

- This video demonstrates how to provide probability bounds in Radiant
- Topics List:
- Use probabilities as input type
- Round up the cutoff value