Probability

M. Drew LaMar
September 7, 2020

“I believe that we do not know anything for certain, but everything probably.”

- Christiaan Huygens

Course Announcements

NO READING ASSIGNMENT FOR Wednesday

Error bars

How to do these in R?

Read and inspect the data.

locustData <- read.csv("../../Datasets/chapter02/chap02f1_2locustSerotonin.csv")
head(locustData)

  serotoninLevel treatmentTime
1            5.3             0
2            4.6             0
3            4.5             0
4            4.3             0
5            4.2             0
6            3.6             0

str(locustData)

'data.frame':   30 obs. of  2 variables:
 $ serotoninLevel: num  5.3 4.6 4.5 4.3 4.2 3.6 3.7 3.3 12.1 18 ...
 $ treatmentTime : int  0 0 0 0 0 0 0 0 0 0 ...

Error bars

First, calculate the statistics by group needed for the error bars: the mean and standard error. Here, tapply is used to obtain each quantity by treatment group.

meanSerotonin <- tapply(locustData$serotoninLevel, 
                        locustData$treatmentTime, 
                        mean)
sdSerotonin <- tapply(locustData$serotoninLevel, 
                      locustData$treatmentTime, 
                      sd)
nSerotonin <- tapply(locustData$serotoninLevel, 
                     locustData$treatmentTime, 
                     length)
seSerotonin <- sdSerotonin / sqrt(nSerotonin)

Error bars

Draw the strip chart and then add the error bars.

\[ \bar{Y} \pm SE_{\bar{Y}} \]

offsetAmount <- 0.2
stripchart(serotoninLevel ~ treatmentTime, 
           data = locustData, 
           method = "jitter", 
           vertical = TRUE)

segments(1:3 + offsetAmount, 
         meanSerotonin - seSerotonin, 
         1:3 + offsetAmount, 
         meanSerotonin + seSerotonin)

points(meanSerotonin ~ c(c(1,2,3) + offsetAmount), 
       pch = 16, 
       cex = 1.2)

Error bars

Draw the strip chart and then add the error bars.

\[ \bar{Y} \pm SE_{\bar{Y}} \]

plot of chunk unnamed-chunk-3

Error bars can mean different things!!!

plot of chunk unnamed-chunk-4

Different error bars!!! \[ \bar{Y} \pm sd \\ \bar{Y} \pm SE_{\bar{Y}} \\ \bar{Y} \pm 2\times SE_{\bar{Y}} \]

Probability Basics

Definition: A random trial is a process or experiment that has two or more possible outcomes whose occurrence cannot be predicted with certainty.

Definition: An event is any potential subset of all the possible outcomes of a random trial.

Definition: The probability of an event is the proportion of times the event would occur if we repeated a random trial over and over again under the same conditions. Probability ranges between zero and one.

Random sampling as a random trial

Instead of events, we have values of random variables.

Parasitic wasps (yuck!): Two categorical variables - Parasitized or not; sex of laid egg (M or F)

The Formulas and Venn Diagrams

Definition: General addition rule \[ \mathrm{Pr[A \ or \ B]} = \mathrm{Pr[A]} + \mathrm{Pr[B]} - \mathrm{Pr[A \ and \ B]} \]

Conditional Probabilities

Definition: The conditional probability of an event is the probability of that event occurring given that another event has already occurred.

Definition: The conditional probability of an event B given that A occurred is \[ \mathrm{Pr[B \ | \ A]} = \frac{\mathrm{Pr[A \ and \ B]}}{\mathrm{Pr[A]}} \]

Definition: General multiplication rule \[ \mathrm{Pr[A \ and \ B]} = \mathrm{Pr[A]}\times\mathrm{Pr[B \ | \ A]} \]

Bayes Rule

Definition: The conditional probability of an event A given that B occurred is \[ \mathrm{Pr[B \ | \ A]} = \frac{\mathrm{Pr[A \ and \ B]}}{\mathrm{Pr[A]}} \]

Definition: General multiplication rule \[ \mathrm{Pr[A \ and \ B]} = \mathrm{Pr[B \ | \ A]}\times\mathrm{Pr[A]} \]

Definition: General multiplication rule \[ \mathrm{Pr[A \ and \ B]} = \mathrm{Pr[A \ | \ B]}\times\mathrm{Pr[B]} \]

Definition: Bayes Rule \[ \mathrm{Pr[B \ | \ A]} = \frac{\mathrm{Pr[A \ | \ B]}\times \mathrm{Pr[B]}}{\mathrm{Pr[A]}} \]

Mutually exclusive vs. independence

Commonly confused!

Definition: Two events are mutually exclusive if they cannot both occur at the same time. \[ \mathrm{Pr[A \ and \ B]} = 0 \]

Definition: Two events are independent if the occurrence of one does not inform us about the probability that the second will occur. \[ \mathrm{Pr[B \ | \ A]} = \mathrm{Pr[B]} \]

Mutually exclusive vs. independence

These two conditions simplify the general additive and multiplicative rules:

If two events are mutually exclusive, then \[ \mathrm{Pr[A \ or \ B]} = \mathrm{Pr[A]} + \mathrm{Pr[B]} \]

If two events are independent, then \[ \mathrm{Pr[A \ and \ B]} = \mathrm{Pr[A]} \times \mathrm{Pr[B]} \]

Visualizing dependency

Independent events

Dependent events

Mosaic plots are awesome!

Totally, dude...

Definition: The probability of an event not occurring is one minus the probability that it occurs. \[ \mathrm{Pr[{\it not}\ A]} = 1-\mbox{Pr[A]} \]

Definition: The law of total probability is given by \[ \begin{align*} \mathrm{Pr[A]} & = \sum_{B\ \mathrm{in} \ \mathcal{M}}\mathrm{Pr[A \ and \ B]} \\ & = \sum_{B\ \mathrm{in} \ \mathcal{M}} \mathrm{Pr[B]}\ \mathrm{Pr[A\ | \ B]}, \end{align*} \] where \( \mathcal{M} \) is a set of mutually exclusive events such that \[ \sum_{B\ \mathrm{in} \ \mathcal{M}}\mathrm{Pr[B]} = 1 \]

Law of total probability and mosaic plots

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Visualizing probability - Probability trees

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Probability distributions

Definition: A probability distribution is a list of the probabilities of all mutually exclusive outcomes of a random trial.

Compare to:

Definition: A probability distribution (or relative frequency distribution) is a list of the probabilities of all values of a random variable in a sample or population.

Discrete probability distributions

alt text

How is this different? same?

Continuous probability distributions

Probability densities alt text

Tips for Solving Probability Problems

Write out the probability that you are being asked to find. Is it a conditional probability? AND? OR?
Identify probabilities that you are given (again, are these conditionals? ANDs? ORs?)
Draw a probability tree (if appropriate)