Probability by Counting: Quick Revision

A brief note on Probability by Counting. Read it before a test. Try each question at the end before you open its answer.

Definition and rules

• Classical: $P(E) = \dfrac{n(E)}{n(S)}$ for equally likely outcomes.
• Axioms: $P(E) \ge 0$, $P(S) = 1$, and $P(A \cup B) = P(A) + P(B)$ for mutually exclusive events.
• Consequences: $P(\varnothing) = 0$, $0 \le P(E) \le 1$, $P(E') = 1 - P(E)$.
• At least one: $1 - P(\text{none})$.
• Odds: odds of $a : b$ in favour give $P = \dfrac{a}{a + b}$.

The addition theorem

• Two events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
• Exactly one: $P(A) + P(B) - 2P(A \cap B)$. Neither: $1 - P(A \cup B)$.
• Limits of the overlap: $\max\{0, P(A) + P(B) - 1\} \le P(A \cap B) \le \min\{P(A), P(B)\}$.
• Three events: singles, minus pairs, plus triple.

Counting tools

• Same style: count the favourable and all outcomes both with order or both without order.
• Selections: $^nC_r$. Arrangements: $^nP_r$ and $n!$.
• Group draw: $\dfrac{^KC_k \times {}^{N-K}C_{r-k}}{^NC_r}$.
• Together: glue into a block and multiply by the arrangements inside.
• Round table: $(n - 1)!$ seatings, and two people are side by side with probability $\dfrac{2}{n - 1}$.

A worked example

A card is drawn from a pack. Find the probability that it is a diamond or a queen.

1. The counts. $13$ diamonds and $4$ queens, and $1$ card is both.
2. The theorem. $\dfrac{13}{52} + \dfrac{4}{52} - \dfrac{1}{52} = \dfrac{16}{52}$.
3. The answer. $\dfrac{4}{13}$.

Practice with new numbers

A program to try

The program checks the addition theorem and the complement rule for three throws of a die, with exact fractions.

Mistakes to avoid

• Unequal outcomes. Counting works only for equally likely outcomes.
• Adding for at least one. Use the complement.
• Forgetting the overlap in the addition theorem.
• Mixing ordered and unordered counts in one fraction.

Check yourself

1. A number is chosen from $1$ to $30$. Find the probability that it is a multiple of $4$.

2. The probability of rain is $0.4$. Find the odds in favour of rain.

3. $P(A) = 0.5$, $P(B) = 0.3$ and $P(A \cup B) = 0.65$. Find $P(A \cap B)$.

4. A committee of $2$ is chosen from $4$ men and $3$ women. Find the probability that it has one man and one woman.

5. Five people sit in a row. Find the probability that two particular people are side by side.

Sources

• NCERT, Mathematics for Class XI (earlier edition), Probability.
• OpenStax, Introductory Statistics, Chapter 3, sections on terminology and on the two basic rules of probability.