Finite events, probability assignments and complements

What is the first step when finding the probability of an event?

Vary q. A and B each contain two outcomes. Compare their probabilities rather than assuming equal likelihood from equal counts

When do A and B have the same probability?

Finite events and exact probability

A finite sample space lists every possible outcome of an experiment. An event is the subset of those outcomes that satisfies a condition, and its probability is found from the appropriate outcome masses

1. 1List the sample space Write every possible outcome once \(\displaystyle S=\{1,2,3,4\}\)
2. 2Identify the event Select the outcomes that satisfy the event condition \(\displaystyle A=\{2,4\}\subseteq S\)
3. 3Use the correct masses Count favourable outcomes only when all outcomes are equiprobable; otherwise add their stated probabilities \(\displaystyle \Pr(A)=\frac{2}{4}=\frac12\quad\text{if equiprobable};\quad\Pr(A)=\Pr(2)+\Pr(4)\)
4. 4Find a complement from one The complement contains all outcomes outside the event \(\displaystyle \Pr(A^{\prime})=1-\Pr(A)\)

Equally likely outcomes and event

\(\displaystyle S=\{1,2,3,4,5,6,7,8\},\quad A=\{2,5,8\}\)

Count the outcomes in A in the complete sample space

Equally likely outcomes and event

\(\displaystyle S=\{1,2,3,4,5,6,7,8\},\quad A=\{2,5,8\}\)

Use the favourable count and the 8 equally likely outcomes to find Pr(A)

Equally likely outcomes and event

\(\displaystyle S=\{1,2,3,4,5,6,7,8\},\quad A=\{2,5,8\}\)

Using the wrong total. One of these lines is right. Which one?

Use a complete sample space

One trial produces one outcome. The sample space contains every possible outcome, and an event is a subset of that space. A singleton event contains one outcome; the event as a set is not the experiment itself

Recover a missing mass before adding an event

A finite experiment selects exactly one of five stated outcomes. The listed masses are probabilities, not counts, and the outcomes need not be equally likely. The complete table must total one

Probability data

\(\displaystyle \frac1{12}+\frac14+\frac16+p+\frac13=1\)

Find the missing probability p.

Probability data

\(\displaystyle \Pr(1)=\frac1{12},\quad\Pr(2)=\frac14,\quad\Pr(3)=\frac16,\quad\Pr(4)=\frac16,\quad\Pr(5)=\frac13\)

How should Pr(A) be found for A={1,3,5} in this unequal distribution?

Probability data

\(\displaystyle \Pr(1)=\frac1{12},\quad\Pr(3)=\frac16,\quad\Pr(5)=\frac13\)

Find the probability of the event shown

Probability data

\(\displaystyle \Pr(A)=\frac7{12}\)

Find Pr(A′), using the complete sample space

Check the complement in two ways

The complement contains outcomes 2 and 4, not another arbitrary two-element set. Adding their masses agrees with subtraction from one

Probability data

\(\displaystyle A=\varnothing\)

What is the probability of the empty event?

Probability data

\(\displaystyle A=S\)

What is the probability of the event containing the entire sample space?

Keep a proposal separate from the actual distribution

A proposed replacement for the probabilities of four outcomes is shown below. Assess this proposal on its own; do not use an invalid proposal to replace a valid table when calculating another event

Probability data

\(\displaystyle -\frac18+\frac38+\frac14+\frac12\)

Find the total of the four proposed entries

Does this proposal define valid probabilities?

Construct the event and complement, ungraded

Use the listed probabilities for outcomes 1, 2, 3 and 4. Write the members of A and A′, calculate both probabilities by adding masses, and verify their sum is one. Explain why counting two outcomes out of four is not enough here. The two component entries do not grade the written sets or explanation

Probability data

\(\displaystyle A=\{2,4\},\quad\Pr(2)=\frac38,\quad\Pr(4)=\frac14\)

Find the event probability

Probability data

\(\displaystyle \Pr(A)=\frac58\)

Find Pr(A′)

Check the sets as well as the masses

The two events are disjoint and together contain the whole sample space. Their probabilities add to one, but their equal numbers of outcomes do not give equal probabilities

Use the full space and the stated probability information

List every event member once, use a count ratio only for equally likely outcomes, and otherwise add their exact masses. A complement is taken relative to the complete sample space. Validate proposed assignments before using them