IP Maths Notes (Lower Sec, Year 1-2): 10) Probability Models

Probability measures uncertainty. This final post develops sample space construction, event notation, and multi-stage probability models.

Learning targets

• Express probabilities as fractions, decimals, or percentages with denominator greater than zero.
• Build sample spaces for single and combined events.
• Draw tree diagrams to track independent and dependent events.
• Apply complementary probability and conditional reasoning.

1. Probability basics

• Probability of event \( A \): \( P(A) = \dfrac{\text{number of favourable outcomes}}{\text{total outcomes}} \).
• Values range from 0 (impossible) to 1 (certain).
• Complement rule: \( P(A') = 1 - P(A) \).

2. Sample spaces

Example: Toss two coins. Outcomes: {HH, HT, TH, TT}. Probability of exactly one head: \( \dfrac{2}{4} = \dfrac{1}{2} \).

3. Tree diagrams

3.1 Independent events

A bag contains 3 red and 2 blue counters. Replace after each draw.

• Probability tree uses same probabilities at each level: \( P(R) = \dfrac{3}{5} \), \( P(B) = \dfrac{2}{5} \).
• Probability of RR: \( \dfrac{3}{5} \times \dfrac{3}{5} = \dfrac{9}{25} \).

3.2 Without replacement (dependent)

If counters are not replaced, second-level probabilities change.

• First draw red, second draw red: \( \dfrac{3}{5} \times \dfrac{2}{4} = \dfrac{3}{10} \).
• First draw blue, second draw red: \( \dfrac{2}{5} \times \dfrac{3}{4} = \dfrac{3}{10} \).

Worked example — Conditional probability

A class has 12 boys and 18 girls. Two students are selected without replacement. Find probability both are girls.

\[ P(\text{GG}) = \dfrac{18}{30} \times \dfrac{17}{29} = \dfrac{306}{870} = \dfrac{51}{145}. \]

4. Complementary reasoning

Sometimes it is easier to calculate the probability of an event not happening.

Example: Probability that at least one six appears in four fair dice rolls.

\[ P(\text{no six}) = \left(\dfrac{5}{6}\right)^4, \quad P(\text{at least one six}) = 1 - \left(\dfrac{5}{6}\right)^4. \]

5. Expected value preview

For a simple game with payouts \( x_i \) and probabilities \( p_i \), expected value \( E(X) = \sum p_i x_i \). Lower-sec exams rarely ask for it directly, but the concept frames decision-making in games of chance.

Try it yourself

1. A box contains 4 green, 3 yellow, and 5 red pens. Two pens are drawn without replacement. Find the probability both are red.
2. The probability of a machine producing a defective item is 0.08. What is the probability that in a batch of three items, at least one is defective?
3. Two dice are rolled. Find the probability the sum is at least 10.
4. A bag contains tickets numbered 1 to 20. One ticket is drawn at random. What is the probability it is a multiple of 3 or 5?

Link back to the overview at https://eclatinstitute.sg/blog/ip-maths-lower-sec-notes/IP-Maths-Lower-Sec-00-Overview for revision sequencing.