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Probability and Conditional Probability
I worked these out from the practice questions. Each code says where to start reading, where to stop, what you can skip, and why it is safe.
1. Read the last sentence first
- Start:
- The very last sentence, to see which group is wanted.
- Stop:
- When you know the wanted group and whether there is a condition.
- Skip:
- The story about why the survey was done.
- Why:
- The question line tells you the top and bottom numbers. Everything else is decoration.
Used in questions 2, 3, 5
2. 'Who' and 'given' shrink the basket
- Start:
- The words 'who', 'given that', 'of those'.
- Stop:
- At the end of that phrase.
- Skip:
- The grand total.
- Why:
- The group named by those words is the only group you divide by. In question 3 and question 5, using the total of everyone gives the tempting wrong answer.
Used in questions 3, 5, 6
3. Mark the bottom number with a finger
- Start:
- The row or column that matches the condition.
- Stop:
- At its total.
- Skip:
- Other rows and columns.
- Why:
- The bottom number is always the total of the group you chose. Put your finger on it before writing the fraction.
Used in questions 2, 3, 5, 7
4. Update the bag when nothing is put back
- Start:
- The phrase 'without replacement' or 'not put back'.
- Stop:
- When you have written the second fraction with one less on the bottom.
- Skip:
- Long decimals; keep fractions.
- Why:
- Both numbers in the second pick change, and fractions let you cancel at the end. Questions 4 and 8 both depend on this.
Used in questions 4, 8
5. Turn probability words into an equation
- Start:
- The given probability, such as 3/5 or 1/3.
- Stop:
- After you write one equation and solve it.
- Skip:
- Trying to guess the missing number.
- Why:
- A probability is a fraction, so it equals an equation. Cross multiply (question 7) or use r(r - 1) (question 8). In Desmos you can type the equation and read the solution.
Used in questions 7, 8
6. Use the overlap rule for 'or'
- Start:
- P(A or B), P(A), P(B).
- Stop:
- When you have found P(A and B).
- Skip:
- Drawing complex diagrams.
- Why:
- P(A and B) = P(A) + P(B) - P(A or B). Then divide by the correct event for the given. Check the order: P(A | B) divides by P(B).
Used in question 6
7. Check the answer size
- Start:
- Your final fraction.
- Stop:
- After one quick test: is it between 0 and 1?
- Skip:
- Re-doing the whole problem.
- Why:
- A probability bigger than 1 or a count instead of a fraction means you used the wrong step. Also 'either order' questions usually double the answer.
Used in questions 1, 4, 8
8. Fill the missing cell before anything else
- Start:
- The row and column totals around the question mark or the letter x.
- Stop:
- When every cell has a number.
- Skip:
- The probability question, until the table is complete.
- Why:
- A table always adds up both ways. One missing cell is found by subtracting from its total, and an unknown x or y is found from the total given in the story (questions 10 and 15).
Used in questions 10, 15
9. Keep only the rows the question names
- Start:
- The phrase that names the group, such as 'Clinic B or Clinic C' or 'who uses the app'.
- Stop:
- When you have crossed out the rows or columns that are not named.
- Skip:
- The totals of the rows you crossed out.
- Why:
- Add the named rows to make your bottom number, then add their matching cells for the top number. In questions 11 and 13 the tempting answers divide by the grand total.
Used in questions 11, 13
10. Turn percents into 1,000 people
- Start:
- The shares and percents in the table.
- Stop:
- When you have counts for every group.
- Skip:
- Fraction rules with percents.
- Why:
- A made-up group of 1,000 turns 60% and 2% into real counts (600 and 12). Then it is the same as a count table, as in question 16. 'At least one' questions are often easier as 1 minus 'none' (question 12).
Used in questions 12, 16