L4 · Probability Clinic

Learning goal

“I can handle a mixed set of two-event probability problems: first decide whether the events are independent or dependent, then build the matching model - full sample space for independent events, shrunken second stage without replacement - and count.”

Sample space — the set of ALL possible outcomes of an experiment. If an outcome is possible but missing from the list, the sample space is wrong.

Independent events — events where the result of one does not change the possible results or the chances of the other.

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.

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

Worked example

Two names are drawn from a hat holding 5 names. The first name is written down and NOT put back before the second draw. Classify, then find the size of the sample space. Step 1: Classify: the first name is kept out, so the second draw's possibilities have changed - dependent. Step 2: Model: ordered pairs, first draw 5 branches, second draw 4 remaining branches. Size: 5 x 4 = 20 outcomes. Step 3: Contrast for safety: if the name HAD been put back, the events would be independent and the space would be 5 x 5 = 25. The classification step decided the denominator before any counting began.

Check yourself — explain it

What single fact about the hat experiment forced the sample space to have 20 outcomes instead of 25?

Paired problem — example still visible

A coin is flipped three times. Classify the events, give the number of endpoints in the tree, and find P(heads, then tails, then heads - in that order). -> Independent (flips never affect each other). Endpoints: 2 x 2 x 2 = 8. The endpoint HTH is 1 of them: P = 1/8.

Faded completion — last step is yours

"A bag holds 4 red marbles and 1 blue marble. Two marbles are drawn without replacement. The sample space has 5 x ___ = ___ ordered outcomes." -> 4; 20

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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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