L2 · Independent Events: Modelling the Sample Space

Learning goal

“I can find the probability of two or more independent events by MODELLING the sample space - with a table, a tree, or a list - and counting: probability = favourable outcomes in the model / total outcomes in the model.”

Independent events — events where the result of one does not change the possible results or the chances of the other (a coin flip never changes a die).

With replacement — a drawn object is put back before the next draw, so the draws are independent and the sample space is rebuilt in full each time.

Tree diagram — a model that branches: each event's outcomes branch off the previous event's outcomes, and each endpoint is one complete outcome.

Area model — a rectangle divided into equal cells (rows x columns); each cell is one combined outcome, so probabilities can be counted as cells.

Worked example

A coin is flipped and a die is rolled. Find P(heads and an even number). Step 1: Model: the 2 x 6 table from L1 - 12 cells, one per ordered pair, all equally likely. Step 2: Mark the favourable cells: the event needs H in the coin AND even in the die: (H,2), (H,4), (H,6) - 3 favourable cells. Step 3: Count, don't guess: P = favourable / total = 3/12 = 1/4. Say it plainly: in this unit a probability is ALWAYS a count from a model. If you cannot point at the favourable outcomes inside a model, you do not have the probability yet.

Check yourself — explain it

Why is the denominator 12 and not 6?

Paired problem — example still visible

Two dice are rolled. Find P(the dice add to 7). -> Build the 6 x 6 table (36 pairs). Pairs summing to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) - 6 favourable. P = 6/36 = 1/6.

Faded completion — last step is yours

"A spinner with 4 equal sections (A, B, C, D) is spun twice. The table has 16 cells. The favourable cells for 'A on both spins' are: only (A,A), so P = 1/___." -> 16

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10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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