L3 · Dependent Events: Two Events Without Replacement

Learning goal

“I can model two dependent events - drawing twice WITHOUT replacement - with a tree or list whose second stage uses the objects that REMAIN, and find probabilities by counting the ordered outcomes in that smaller sample space.”

Dependent events — events where the first event changes what the second event can produce, so the second event's sample space (and its chances) are different.

Without replacement — a drawn object is kept out; it cannot be drawn again. This is what makes two draws dependent.

Ordered pair — an outcome written (first result, second result). (Red, Blue) and (Blue, Red) are different outcomes when the draws happen in order.

Remaining outcomes — the outcomes still available at the second stage, after the first outcome has been removed from the set.

Worked example

A bag holds 4 letter tiles: A, B, E, D. Two tiles are drawn WITHOUT replacement. Find P(both tiles are vowels). Step 1: The vowels are A and E. Label everything - each tile is distinct already. Step 2: Build the model. First draw: 4 branches (A, B, E, D). From EACH first-draw branch, the second draw has only 3 branches - the tile just drawn is gone. For example, if A was drawn first, the second draw branches to B, E, D (no A: you cannot draw the same tile twice). Step 3: Total ordered outcomes: 4 x 3 = 12: (A,B), (A,E), (A,D), (B,A), (B,E), (B,D), (E,A), (E,B), (E,D), (D,A), (D,B), (D,E). Notice there is no (A,A) - the diagonal pairs from the with-replacement table are gone. Step 4: Favourable (both vowels): (A,E) and (E,A) - 2 outcomes. P = 2/12 = 1/6. Say it plainly: dependent events change the MODEL, not just the numbers: the second stage has fewer branches, and the "same object twice" outcomes disappear.

Check yourself — explain it

Why is (A,A) not in this sample space?

Paired problem — example still visible

Five students - Ana, Bo, Cy, Di, Ed - are in a draw. A winner is chosen, then a runner-up is chosen from the remaining students (no one can hold both spots). How many ordered outcomes are there, and what is P(Ana wins and Bo is runner-up)? -> Outcomes: 5 x 4 = 20. The outcome (Ana, Bo) is exactly 1 of them: P = 1/20.

Faded completion — last step is yours

"Numbers 1, 2, 3, 4, 5 are on cards. Two cards are drawn without replacement. The sample space has 5 x ___ = ___ ordered outcomes." -> 4; 20

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