Grade 10 · Restrict a conditional sample space · 534 of 750
Restrict a conditional sample space · practice 57
A chosen group contains 4 successes and 8 failures; 10 cases lie outside it. Find P(success | chosen group) and the fraction of all cases in the chosen group. Follow the calculation, then test the quantities in the animated example.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
A chosen group contains 4 successes and 8 failures; 10 cases lie outside it. Find P(success | chosen group) and the fraction of all cases in the chosen group.
Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.
A starting point
The conditional denominator excludes outside cases; the group-membership denominator includes them.
Work through the reasoning
Step 1
Translate the givens
The chosen group contains 4 + 8 = 12 cases. Conditioning restricts attention to these cases.
Step 2
Calculate with the model
Of these, 4 succeed: P(success | group) = 4/12 ≈ 33.33333%.
Step 3
Check and interpret
The whole collection has 12 + 10 = 22 cases. Its chosen-group fraction is 12/22 ≈ 54.54545%.
The answer
P(success | group) = 4/12; P(group) = 12/22.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
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The mathematical idea
Conditioning restricts the sample space to cases meeting a specified condition. The denominator becomes the size of that chosen group, rather than the size of the entire collection. Counts outside the group disappear from the conditional calculation even when they were visible initially. Starting quantities: Successes in chosen group = 4; Failures in chosen group = 8; Cases outside chosen group = 10.
P(success | chosen group) = successes / (successes + failures)
03 · Reflect and transfer
Explain what changes and why.
Why can changing the outside count change group membership probability without changing the conditional success probability?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.