Study guide: Count Carefully in Probability Models
Use two-way counts, conditional sample spaces, independence tests, and updated draws.
Ideas to explain in your own words
- Build a two-way table and distinguish a joint event from a total category.
- Restrict the sample space before calculating a conditional probability.
- Compare a conditional probability with its unconditional probability.
- Multiply a first-draw probability by the correctly updated second-draw probability.
Watch your reasoning
- The blue-circle cell contains six cards. The sample space is all 24 cards, so P(blue and circle) = 6/24 = 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.
- P(red given circle) = 6/(6 + 2) = 3/4, while P(red) = 12/24 = 1/2. Knowing circle changes the red probability, so the events are dependent in this model.
- First red is 4/6. After it is removed, three red cards remain among five, so the chance is (4/6)(3/5) = 12/30 = 2/5.
A worked review example
Read Joint and Marginal Counts: P(red and circle) = 4/24 = 1/6. P(circle) = (4 + 6)/24 = 10/24 = 5/12. The latter includes both colors in the circle column. Find a probability after information is given: 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. Test Whether Two Attributes Are Independent: P(red) = 1/2 and P(red given circle) = 1/2, so red and circle are independent here. The joint check gives 1/6 = (1/2)(1/3). Track Two Draws Without Replacement: P(red then blue) = (3/5)(2/4) = 6/20 = 3/10. The second probability is conditioned on red having been removed.
Make a practice plan
- Try the supplied lesson problems without looking at the answers.
- Compare your method with the worked examples and explain any correction.
- Use the new review cases, then complete the transfer task on paper or in your own document.