Counting & Probability (Ex 10D)

Year 11 Mathematical Methods · Cambridge Methods 1&2, Counting Methods · Mr Wong

Name

Class

Date

Learning intentions.

Count total equally-likely outcomes for an arrangement or selection
Count favourable outcomes using restrictions, "exactly" and "at least"
Write each probability as an exact, simplified fraction

Key results

For equally-likely outcomes $P(\text{event})=\dfrac{n(\text{favourable})}{n(\text{total})}$. Count both with the multiplication principle, $^nP_r$ or $^nC_r$, then simplify. "At least one" is often quickest as $1-P(\text{none})$.

Warm-up — set up the fraction

1. The $5$ distinct letters of MATHS are arranged at random.

a Total number of arrangements

b $P(\text{the word ends in }S)$

c $P(\text{the word starts with }M)$

Part A — Probability with arrangements

2. The digits $1,2,3,4,5$ are arranged at random to form a five-digit number.

a $P(\text{the number is even})$

b $P(\text{the number is greater than }30\,000)$

3. Four people line up at random. Find $P(\text{two particular people }A\text{ and }B\text{ stand together})$.

4. Four boys and three girls stand in a row at random. Find $P(\text{all four boys stand together})$. (Glue the boys into a block.)

Counting 10D — continued

Part B (selections) · Challenge

Name

Part B — Probability with selections

5. A committee of $3$ is chosen at random from $5$ men and $3$ women.

a Total number of committees

b $P(\text{all three are women})$

c $P(\text{exactly two are women})$

d $P(\text{at least one woman})$

6. Three students are chosen at random from a class of $10$, of whom $4$ are left-handed. Find $P(\text{all three are left-handed})$.

7. Two counters are drawn at random from a bag containing $3$ red and $4$ blue counters. Find $P(\text{both counters are red})$.

Challenge

8. A committee of $4$ is chosen at random from $9$ students, one of whom is Priya.

a Find $P(\text{Priya is on the committee})$.

b Explain how parts (a) relates to choosing the other $3$ members.