H2 Maths Notes (JC 1-2): 6.1) Probability

Study cadence\ Tighten your language: every time you quote a probability, state the event clearly first. Keep a running list of assumptions (mutually exclusive, independent, equally likely) before applying shortcuts.

Core Definitions

• Sample space ( S ) contains all mutually exclusive outcomes.
• Event ( A \subseteq S ) has probability ( P(A) = ) number of favourable outcomes divided by number of total outcomes (for equally likely cases) or the frequency assigned by the model.
• Complement: ( P(A^c) = 1 - P(A) ).
• Addition law: ( P(A \cup B) = P(A) + P(B) - P(A \cap B) ).
• Conditional probability: ( P(A \mid B) = \frac{P(A \cap B)}{P(B)} ) whenever ( P(B) > 0 ).
• Independence: ( A ) and ( B ) are independent if ( P(A \cap B) = P(A) P(B) ); equivalently ( P(A \mid B) = P(A) ) and ( P(B \mid A) = P(B) ).

Diagram Workflows

Venn diagrams

1. Label each region with algebraic expressions (e.g. ( x ) for the overlap, ( 0.4 - x ) for the remaining part of ( A )).
2. Use the total probability constraint (sum of all regions equals 1) to solve for the unknowns.
3. When three sets appear, build the diagram region by region, starting from the three-way overlap.

Example -- Scholarship shortlisting

Let ( A ) be "strong portfolio" and ( B ) be "strong interview". Given ( P(A) = 0.65 ), ( P(B) = 0.55 ), and ( P(A \cap B) = 0.42 ):

• ( P(A^c \cap B) = 0.55 - 0.42 = 0.13 ).
• ( P(A \cap B^c) = 0.65 - 0.42 = 0.23 ).
• ( P(A^c \cap B^c) = 1 - 0.65 - 0.55 + 0.42 = 0.22 ).
• ( P(A \mid B) = \frac{0.42}{0.55} \approx 0.764 ).

Tree diagrams

1. Multiply along branches to obtain joint probabilities (e.g. ( P(A \cap B) = P(A) P(B \mid A) )).
2. Sum the relevant final nodes to find the total probability of an event.
3. When conditional information is provided about the second stage, write it beside the branches before multiplying.

Example -- Quality control

A factory inspects items with two stages:

• Stage 1: pass with probability 0.9.
• Stage 2: conditional pass probability is 0.95 if Stage 1 passed, 0.6 if Stage 1 failed.

Compute the probability of an item passing both stages:

1. Branch values: pass then pass has probability ( 0.9 \times 0.95 = 0.855 ).
2. If the second stage pass probability is to be found given the item passed Stage 2, apply Bayes:
   • Total Stage 2 pass probability = ( 0.9 \times 0.95 + 0.1 \times 0.6 = 0.915 ).
   • Probability an item passed Stage 1 given it passed Stage 2 = ( \frac{0.855}{0.915} \approx 0.935 ).

Common Question Types

Showing independence (or dependence)

• Compute both ( P(A \cap B) ) and ( P(A) P(B) ); equality means independent, inequality means dependent.
• Alternatively, compare ( P(A \mid B) ) with ( P(A) ).

Conditional complements

• ( P(A^c \mid B) = 1 - P(A \mid B) ).
• When the question gives the probability of "at least one" event happening, convert to the complement of "none".

Total probability rule

For a partition ( B_1, B_2, \dots, B_k ) of the sample space:

[ P(A) = \sum_{i=1}^{k} P(A \mid B_i) P(B_i). ]

In H2 questions this often appears in genetics or reliability contexts.

Calculator and Notation Tips

• Casio: use the TREE mode to visualise multi-stage events; store intermediate probabilities in variables A, B, C if you need to reuse them across branches.
• TI: use lists to capture outcomes and apply cumSum to compute running totals for "at least" or "at most" questions.
• Always define events in words before introducing symbols; MOE marking schemes penalise answers that show algebra without context.

Exam Watch Points

• State assumptions (mutually exclusive or independent) explicitly before cancelling terms.
• For sequential processes, highlight whether sampling is with or without replacement; it changes conditional probabilities immediately.
• When using Bayes, write out the denominator expansion so the marker sees the full weighting.
• Double-check rounding: most questions accept 3 significant figures unless stated otherwise.

Quick Revision Checklist

• [ ] Convert word problems into formal event notation consistently.
• [ ] Draw and label Venn or tree diagrams before solving algebraically.
• [ ] Execute conditional probability calculations without skipping intermediate steps.
• [ ] Distinguish between mutually exclusive and independent events (they are not the same).
• [ ] Apply total probability and Bayes reliably under time pressure.

Next steps: practise 2-by-2 Bayes table questions (medical testing, customer segmentation) and extend into the discrete random variable toolkit covered in Topic 6.2.