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1 of 8 0-day streak

Multi-stage probability, multiplication and total probability

Question 1 of 8

No calculator

A trial selects one block from a collection at each stage

The initial category counts, in the branch order shown, are round = 9, square = 5, flat = 6

At each stage, choose uniformly at random from the individual items currently in the container

The trial has 2 stages: stage 1 and stage 2. After each selection, the item is returned before the next stage

The ordered terminal outcomes form the sample space \(S\). Let \(E\) be the event that the ordered outcome is (square, flat)

A directed probability tree with 13 nodes. Its branches are Start to round labelled 9/20; Start to square labelled ?; Start to flat labelled 3/10; round to (round, round) labelled 9/20; round to (round, square) labelled 1/4; round to (round, flat) labelled 3/10; square to (square, round) labelled 9/20; square to (square, square) labelled 1/4; square to (square, flat) labelled 3/10; flat to (flat, round) labelled 9/20; flat to (flat, square) labelled 1/4; flat to (flat, flat) labelled 3/10.9/20?3/109/201/43/109/201/43/109/201/43/10Startroundsquareflat(round, round)(round, square)(round, flat)(square, round)(square, square)(square, flat)(flat, round)(flat, square)(flat, flat)

If needed, scroll sideways to follow each complete path

Read stages from left to right. A ? marks a branch probability to calculate. Terminal labels show the complete ordered outcomes
(a)1 mark

Which calculation plan matches the probability of the required path and event?

(b)2 marks

Fill the labelled branch and terminal-path probabilities exactly

(c)1 mark

Find the exact probability of the ordered path \(\mathrm{square},\, \mathrm{flat}\)

Format: exact fraction or terminating decimal