Handling Data 7
3 marks · NormalA spinner can land on red, blue, or green. The probability of red is 0.3, the probability of blue is 0.4, and the probability of green is 0.3. Two independent spins are made.
Explain, with reference to a tree diagram, why the sum of the probabilities of all the possible outcomes is equal to 1.
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
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Each possible outcome probability is found by multiplying the probabilities for each stage. The probabilities for all possible outcomes add up to 1 because the spinner must land on one of the colours each time, and these outcomes cover all possible combinations in two spins.
Must show: Multiplying for branches, List of all possible outcomes, Statement that total probability = 1
Also accept: Explanation mentioning exhaustive and mutually exclusive outcomes
Common mistakes: Omitting to mention all outcomes; Not explaining multiplication at branches
Show mark scheme and worked solution
Marking points
M1 · 1 mark · Explains that at each branch, probabilities for red, blue, green add to 1 and are consistent for both spins.
M2 · 1 mark · States that each possible outcome probability is found by multiplying the probabilities of the colours on each spin.
A1 · 1 mark · Clearly reasons (or shows by calculation) that the sum of all combined outcome probabilities is 1 because all possible combinations are included.
Worked solution
Each spin is independent. For the first spin, the outcomes are red (0.3), blue (0.4), and green (0.3).
The second spin branches from each of these, again with 0.3, 0.4, 0.3.
There are 9 possible outcomes (red-red, red-blue, red-green, blue-red, ..., green-green).
Each combined outcome probability is found by multiplying the two probabilities. For example, P(red then blue) = 0.3 × 0.4 = 0.12.
Adding up the probabilities for all 9 outcomes: (0.3 + 0.4 + 0.3) × (0.3 + 0.4 + 0.3) = 1 × 1 = 1.
Therefore, the probabilities of all possible outcomes together make 1, which always happens because all possible outcomes are represented in the tree diagram and these are mutually exclusive and exhaustive.