Mathematics 9709 · Probability & Statistics 1

Probability basics

Probability is a number from $0$ to $1$ .
If all outcomes are equally likely, use winning outcomes over all outcomes.
If the question asks for an expected number, multiply the probability by the number of trials.

Count the outcomes that win

An outcome is one result, like rolling a $5$ .
An event is a set of outcomes, like rolling an odd number: $\{1,3,5\}$ .
Words such as fair, random, and equally likely are your permission to count outcomes.

$P(\text{event})=\dfrac{\text{outcomes that win}}{\text{all equally likely outcomes}}$

Worked example

First rung: one fair die

Odd numbers on a die are $1,3,5$ , so there are 3 winning outcomes.
A fair die has 6 equally likely outcomes.

Answer

P(\text{odd})=\dfrac36=\dfrac12

Not A means everything outside A, so $P(A')=1-P(A)$ .
At least one often means use the back door: $1-P(\text{none})$ .
Expected number means probability as a rate: $\text{expected count}=nP(A)$ .

Worked example

One extra step: use the complement

A fair die is rolled. Find the probability of not getting a 6.

The one losing outcome is 6, so $P(6)=\tfrac16$ .
Everything else is $1-\tfrac16=\tfrac56$ .

Answer

\dfrac56

Worked example

Expected count, not just a probability

The probability of rain on a day is $0.2$ . Find the expected number of dry days in a year.

Dry is the complement: $P(\text{dry})=1-0.2=0.8$ .
Turn the rate into a count: $365\times0.8=292$ .

Answer

292\text{ dry days}

Common mistake

Do not count outcomes unless they are equally likely. The sums from two dice are not equally likely. The 36 ordered pairs are.

The fast check

A probability bigger than $1$ is impossible.
A probability below $0$ is impossible.
An expected count can be a decimal, because it is an average over many repeats.

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Probability basics

Probability is a number from $0$ to $1$ .
If all outcomes are equally likely, use winning outcomes over all outcomes.
If the question asks for an expected number, multiply the probability by the number of trials.

Count the outcomes that win

An outcome is one result, like rolling a $5$ .
An event is a set of outcomes, like rolling an odd number: $\{1,3,5\}$ .
Words such as fair, random, and equally likely are your permission to count outcomes.

$P(\text{event})=\dfrac{\text{outcomes that win}}{\text{all equally likely outcomes}}$

Worked example

First rung: one fair die

Odd numbers on a die are $1,3,5$ , so there are 3 winning outcomes.
A fair die has 6 equally likely outcomes.

Answer

P(\text{odd})=\dfrac36=\dfrac12

Not A means everything outside A, so $P(A')=1-P(A)$ .
At least one often means use the back door: $1-P(\text{none})$ .
Expected number means probability as a rate: $\text{expected count}=nP(A)$ .

Worked example

One extra step: use the complement

A fair die is rolled. Find the probability of not getting a 6.

The one losing outcome is 6, so $P(6)=\tfrac16$ .
Everything else is $1-\tfrac16=\tfrac56$ .

Answer

\dfrac56

Worked example

Expected count, not just a probability

The probability of rain on a day is $0.2$ . Find the expected number of dry days in a year.

Dry is the complement: $P(\text{dry})=1-0.2=0.8$ .
Turn the rate into a count: $365\times0.8=292$ .

Answer

292\text{ dry days}

Common mistake

Do not count outcomes unless they are equally likely. The sums from two dice are not equally likely. The 36 ordered pairs are.

The fast check

A probability bigger than $1$ is impossible.
A probability below $0$ is impossible.
An expected count can be a decimal, because it is an average over many repeats.