Long-Run Chance and the Addition Rule: Quick Revision

A brief note on Long-Run Chance and the Addition Rule. Read it before a test. Try each question at the end before you open its answer.

Two ways to find a probability

• From data: relative frequency $= \dfrac{\text{times it happened}}{\text{tries}}$.
• From equal chances: $P(A) = \dfrac{\text{outcomes in } A}{\text{all outcomes}}$.
• A model: every probability is between $0$ and $1$, and all of them add up to $1$.

The long run

• Law of large numbers: the share of successes gets close to the probability as the tries grow.
• The band: in about $95$ runs out of $100$, the share is within $2\sqrt{p(1-p)/n}$ of $p$.
• A fair coin: the half-width is $\dfrac{1}{\sqrt{n}}$.
• Only the share settles: the count of heads minus half does not have to shrink.

Complements and the addition rule

• Complement: $P(\text{not } A) = 1 - P(A)$.
• Exclusive events: $P(A \cup B) = P(A) + P(B)$.
• Any two events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

A worked example

A made-up shop records $80$ shoppers. $44$ pay by card, $30$ pay by phone and $14$ use both on different visits.

1. Card or phone. $44 + 30 - 14 = 60$ shoppers, so the probability is $\dfrac{60}{80} = 0.75$.
2. Neither. $1 - 0.75 = 0.25$, which is $20$ shoppers.
3. Card only. $44 - 14 = 30$ shoppers, so the probability is $\dfrac{30}{80} = 0.375$.

Practice with new numbers

A program to try

The program checks the worked example, and then picks a shopper at random $10000$ times to watch the share settle.

Mistakes to avoid

• Counting the overlap twice. Subtract $P(A \cap B)$ once.
• Expecting a short run to match the probability. Only a long run settles.
• Thinking results even out. The share settles, but the count does not catch up.
• A model that does not add up to $1$. Look for a missing outcome.

Check yourself

1. A made-up machine jammed $14$ times in $350$ runs. Estimate the probability of a jam.

2. A fair coin is tossed $900$ times. Give the band for the share of heads.

3. $P(A) = 0.3$, $P(B) = 0.5$ and $P(A \cap B) = 0.1$. Find $P(A \cup B)$.

4. Two events are mutually exclusive, with probabilities $0.25$ and $0.4$. Find the probability of neither.

Sources

• OpenStax, Introductory Statistics, Chapter 3, the sections on terminology, the basic rules of probability and Venn diagrams.