Two-Way Tables and Fair Games: Quick Revision

A brief note on Two-Way Tables and Fair Games. Read it before a test. Try each question at the end before you open its answer.

Conditional probability

• From a table: $P(A \mid B)$ is the count of $A$ and $B$ divided by the count of $B$.
• From shares: $P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$.
• Order matters: $P(A \mid B)$ and $P(B \mid A)$ have the same top and different bottoms.

Independence in a table

• Same share: the share of $A$ is the same in every group of $B$, so $P(A \mid B) = P(A)$.
• Product rule: $P(A \cap B) = P(A) \times P(B)$.
• Expected count: $\dfrac{\text{row total} \times \text{column total}}{\text{grand total}}$.
• Real samples: small differences can be chance. A link is not a cause.

Expected value

• Rule: $E = x_1 p_1 + x_2 p_2 + \dots$, the long-run average.
• Gain: expected prize minus the cost. A fair game has an expected gain of $0$.
• House edge: a bet of $1$ token on red on a wheel with $37$ pockets has $E = -\dfrac{1}{37}$ tokens.

A worked example

A made-up survey of $100$ people: $30$ drink tea, $60$ drink coffee, and $18$ drink both.

1. Coffee given tea. $\dfrac{18}{30} = 0.6$.
2. Coffee given no tea. $\dfrac{60 - 18}{70} = \dfrac{42}{70} = 0.6$.
3. Independent? Yes. Both shares equal $P(\text{coffee}) = 0.6$, and the expected count $\dfrac{30 \times 60}{100} = 18$ matches.

Practice with new numbers

A program to try

The program checks the worked example with exact fractions.

Mistakes to avoid

• Dividing by the grand total for a conditional probability. Divide by the group you are told about.
• Swapping the condition. Read the question twice: what is known, and what is asked?
• Forgetting the cost of a game. The expected prize is not the gain.
• Calling an association a cause. A table shows a link only.

Check yourself

1. Of $50$ made-up members of a club, $20$ are adults, and $15$ of the adults play chess. Find P(chess given adult).

2. $P(A) = 0.4$, $P(B) = 0.5$ and $P(A \cap B) = 0.2$. Are $A$ and $B$ independent?

3. A game pays $5$ tokens with probability $0.3$ and costs $2$ tokens. Find the expected gain.

4. What prize, paid with probability $0.25$, makes a $3$ token game fair?

Sources

• OpenStax, Introductory Statistics, Chapter 3 (contingency tables, independent events) and Chapter 4 (expected value).