Probability from data and from equal chances

What you will be able to do

• find a probability from a record of what happened, as a relative frequency;
• find a probability by counting equally likely outcomes;
• check that a probability model has every value between $0$ and $1$ and a total of $1$.

The late bus

Here is a made-up record from one school. Over $75$ school days, the bus was late on $21$ days. What is the chance that it is late tomorrow?

A bus is not a die. Nothing about its shape makes "late" and "on time" equally likely. But we have data. The bus was late on this share of the days:

This share is the relative frequency of a late bus. It is the number of times the event happened, divided by the number of tries. A relative frequency is our best estimate of the probability when we cannot count equal chances.

Equal chances

Some experiments have outcomes that are equally likely because of their shape. A fair coin, a fair die, a spinner with equal sectors and a well-shuffled pack of cards are like this. The list of all the outcomes is the sample space. An event is a set of outcomes from the sample space. When the outcomes are equally likely,

For a fair die, the event "more than $4$" holds the outcomes $5$ and $6$. So $P(\text{more than } 4) = \dfrac{2}{6} = \dfrac{1}{3}$.

A spinner needs one more idea. If the pointer is equally likely to stop at any angle, a sector's probability is its angle divided by $360$ degrees. A sector of $90$ degrees has probability $\dfrac{90}{360} = 0.25$.

Which way fits?

• Equal chances fit coins, dice, spinners, shuffled cards and names drawn from a hat.
• A relative frequency fits buses, weather, a drawing pin, a seed that may sprout and a new phone that may break.
• A third kind is a judgement, such as "I think there is a $70\%$ chance that our team wins". It can be useful, but two people may give different numbers.

A probability model

A made-up survey asks $250$ students how they come to school.

The bottom row is a probability model: a list of the outcomes with a probability for each one. Two rules hold for every model. Each probability is between $0$ and $1$. The probabilities add up to $1$, because exactly one outcome happens. Here $0.26 + 0.4 + 0.18 + 0.16 = 1$.

Try it in Python

The program counts the late days in a short made-up record. Then it finds a probability for a die in two ways: by counting the faces, and by rolling the die $600$ times on the computer.

Your turn

The usual mistakes

• Treating outcomes as equally likely when they are not. A bus is late or on time, but that does not make each chance $\dfrac{1}{2}$.
• Dividing by the wrong total. A relative frequency divides by the number of tries, not by the number of failures.
• A model that does not add up. If the probabilities of all the outcomes do not add up to $1$, an outcome is missing or a number is wrong.
• Trusting a tiny record. Three late days out of five tells you much less than $21$ out of $75$.

Remember these

• Relative frequency: the number of times an event happened, divided by the number of tries.
• Equal chances: $P(A)$ is the number of outcomes in $A$ divided by the number of all outcomes.
• Every probability is between $0$ and $1$, and the probabilities of all the outcomes add up to $1$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Data from a made-up weather station show rain on $36$ of $120$ days in one season. Estimate the probability of rain on a day of that season.

2. A fair ten-sided die has the numbers $1$ to $10$. What is the probability of a prime number?

3. A spinner has three sectors of $90$, $120$ and $150$ degrees. What is the probability that it stops in the largest sector?

4. An experiment has only three outcomes, A, B and C. If $P(A) = 0.5$ and $P(B) = 0.3$, what is $P(C)$?

5. You want the probability that a new kind of phone battery lasts more than two years. Do you count equal chances or use data? Why?

References

• OpenStax, Introductory Statistics, Chapter 3, the section on terminology.