Find a probability after information is given
Restrict the sample space before calculating a conditional probability.
Supplied materials
- A fictional box contains 20 cards: 8 red and 12 blue. Of the red cards, 6 have a circle; of the blue cards, 3 have a circle. Each card is equally likely to be drawn.
- No physical cards are needed. Circle and color are separate attributes.
Learn the idea
A conditional probability asks for a chance given some information. That information restricts the set of possible outcomes. The denominator counts only outcomes allowed by the condition.
“Red given circle” and “circle given red” are different questions. Their numerators can match while their denominators differ. State the condition before doing any division.
- Make a two-way table: red circle 6, red not-circle 2, blue circle 3, blue not-circle 9.
- Check that all four counts total 20.
- For red given circle, count all circle cards before finding the fraction.
- For the question below, restrict the set to blue cards first.
See the method in action
There are 6 + 3 = 9 circle cards. Of these, 6 are red, so P(red given circle) = 6/9 = 2/3. P(red) before learning the shape is 8/20 = 2/5, a different probability.
Check your reasoning
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See the answer and explanation
1/4
The condition restricts the set to 12 blue cards, of which 3 have a circle. The probability is 3/12 = 1/4. Using all 20 cards ignores the supplied condition.
Take it further
Compare P(circle given red) = 6/8 = 3/4 with P(red and circle) = 6/20 = 3/10. Explain the different denominators.
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Longer courses for learners ages 14+
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