13.2 Probability and Conditional Probability
Key Takeaways
- When outcomes are equally likely, P(event) = (favorable outcomes) / (total outcomes), a number from 0 to 1.
- A two-way table stores joint counts in the interior cells and totals in the margins; check that rows, columns, and the grand total agree before you divide.
- Conditional probability ('given,' 'among,' 'if') uses only the matching row or column as the new total, not the grand total.
- P(A given B) is not automatically P(B given A); the denominator changes when the condition changes.
- Two events are independent if knowing one does not change the probability of the other; if the conditional probability differs from the ordinary probability, the events are dependent in that table.
Probability on the PSAT 8/9 is the chance of an outcome, written as a number from 0 (impossible) to 1 (certain). Almost every item is a counting problem: P(event) = (number of favorable outcomes) / (number of possible outcomes), when the outcomes are equally likely.
You already use fractions in Ratios, Rates, Proportions, and Units. Probability is the same fraction idea with a chance meaning. Percents that appear in probability stories convert the same way as in Percentages: 9/20 = 0.45 = 45%.
Equally likely outcomes
Example. A spinner is split into 8 equal sectors: 3 red, 2 blue, 3 green. The spinner is fair, so each sector has probability 1/8.
P(red) = 3/8
P(not red) = 1 − 3/8 = 5/8, which matches the 2 blue + 3 green sectors. The complement rule P(not A) = 1 − P(A) is often faster than recounting.
Example. A bag holds 4 red, 3 blue, and 5 green marbles (12 marbles). One marble is drawn at random.
P(green) = 5/12
P(red or blue) = (4 + 3)/12 = 7/12
If a question says the marble is replaced and the bag is shuffled, a second draw still has 12 marbles. If the marble is kept out, the total drops to 11 and the color counts change. That is the grade 8–9 picture of dependent draws: the first result changes the next fraction.
A probability of 0 means the event cannot happen with the listed outcomes (drawing a yellow marble from a bag that has none). A probability of 1 means the event must happen (drawing a marble that is red, blue, or green from that 12-marble bag).
Two-way tables
Many PSAT 8/9 probability questions hide in a two-way table (a frequency table with two categories). One category labels rows; the other labels columns. Interior cells are joint counts. Margin totals are row or column sums.
Worked table. Forty students are classified by grade and by whether they take band.
| Band | No band | Total | |
|---|---|---|---|
| 8th grade | 9 | 11 | 20 |
| 9th grade | 6 | 14 | 20 |
| Total | 15 | 25 | 40 |
Check the arithmetic before you compute probabilities:
- 9 + 11 = 20 and 6 + 14 = 20 (row totals)
- 9 + 6 = 15 and 11 + 14 = 25 (column totals)
- 20 + 20 = 40 and 15 + 25 = 40 (grand total)
If a table is internally inconsistent, you misread a cell. On a well-written item the totals agree.
Ordinary (unconditional) probability
A student is selected at random from all 40.
P(band) = 15/40 = 3/8 = 0.375
P(8th grade) = 20/40 = 1/2
P(8th grade and band) = 9/40 = 0.225
The last one is the interior cell over the grand total. 'And' means the overlap box.
P(8th grade or band) uses the inclusive-or count: 8th graders plus band members minus the students counted twice (8th-grade band).
9 + 11 + 6 = 26, or 20 + 15 − 9 = 26.
P(8th or band) = 26/40 = 13/20 = 0.65.
The 14 ninth graders not in band are the only students left out: 40 − 14 = 26. Same count.
Conditional probability: restrict the total
Conditional probability is probability given extra information. In a two-way table, 'given' almost always means: throw away every row or column that does not match the condition, then form a new fraction.
Language to listen for:
- 'given that the student is an 8th grader'
- 'among the 8th graders'
- 'if the student is in band'
- 'a band student is selected at random'
Example. A student is selected at random from the 8th graders. What is the probability the student is in band?
The condition is 8th grade, so the new total is 20, not 40. Favorable: the 9 eighth graders in band.
P(band | 8th grade) = 9/20 = 0.45
The vertical bar in the notation is read 'given.' In plain language: of the 20 eighth graders, 9 are in band, so the chance is 9 out of 20. You need that restricted total of 20, not the original 40.
Example. A student is selected at random from those in band. What is the probability the student is an 8th grader?
Now the total is the band column: 15. Favorable: 9.
P(8th grade | band) = 9/15 = 3/5 = 0.6
9/20 and 9/15 are different. That is the whole point of 'given.' Students who keep the denominator 40 on a 'given' question are answering a different problem — the ordinary joint probability 9/40.
If you ever reverse the story, rewrite it in words first: 'among eighth graders, who is in band?' versus 'among band members, who is an eighth grader?' Those are two different restricted groups.
A second table, with a science flavor
A biology class records 50 leaf samples as sun or shade and as thick or thin.
| Thick | Thin | Total | |
|---|---|---|---|
| Sun | 18 | 7 | 25 |
| Shade | 10 | 15 | 25 |
| Total | 28 | 22 | 50 |
P(thick) = 28/50 = 14/25 = 0.56
P(thick | sun) = 18/25 = 0.72
P(thick | shade) = 10/25 = 0.40
P(sun | thick) = 18/28 = 9/14 ≈ 0.643
Sun leaves in this table are more often thick than shade leaves. The condition changes the probability, so thickness and light are not independent in this data set.
Independent vs. dependent at 8/9 level
Two events are independent if knowing that one happened does not change the probability of the other. In symbols, P(A given B) equals P(A). Two events are dependent if the conditional probability does change.
You do not need a long independence proof. Compare two fractions from the table.
Independent example. A cafeteria survey of 50 students:
| Juice | Water | Total | |
|---|---|---|---|
| Walk | 12 | 8 | 20 |
| Bus | 18 | 12 | 30 |
| Total | 30 | 20 | 50 |
P(juice) = 30/50 = 0.6
P(juice | walk) = 12/20 = 0.6
P(juice | bus) = 18/30 = 0.6
The juice probability is 0.6 whether or not you know the student walks. In this table, drink choice and how the student gets to school are independent.
Check the product rule, which is the same idea: P(walk and juice) should equal P(walk) × P(juice) when the events are independent.
P(walk) = 20/50 = 0.4
P(walk) × P(juice) = 0.4 × 0.6 = 0.24
P(walk and juice) = 12/50 = 0.24
The match confirms independence for these two categories.
Dependent example. Return to the band table. P(band) = 15/40 = 0.375, but P(band | 8th) = 9/20 = 0.45. Those are not equal, so grade and band are dependent in that class. Knowing the grade does change the chance of band.
Everyday pictures:
- Independent: two fair coin flips; spinning a spinner twice; drawing a card, replacing it, and drawing again.
- Dependent: drawing two marbles without replacement; most two-way tables whose row percentages differ.
Replacement is the physical version of 'the first result does not change the next fraction.' Without replacement, the bag is different on the second draw, so the events are dependent.
Trap list
- Wrong denominator. 'Given 8th grade' means divide by the 8th-grade total, not by 40.
- Flipping the condition. P(band | 8th) is not P(8th | band).
- Adding interiors that overlap without subtracting the students who sit in both categories.
- Assuming independence because two events can both happen. 'Can both happen' is not the same as 'do not affect each other.'
- Percent vs. probability. 9/20 is 45%, which is 0.45 as a probability. If the item wants a probability, 45 without a percent is the wrong scale; if it wants a percent, 0.45 is unfinished.
A spinner is divided into 10 equal sectors: 4 green, 3 blue, 2 yellow, and 1 red. What is the probability that the spinner lands on yellow?
A school surveys 40 students, with 9 eighth graders in band, 11 eighth graders not in band, 6 ninth graders in band, and 14 ninth graders not in band. A student is selected at random from the eighth graders. What is the probability the student is in band?
Two events A and B are independent. Which statement must be true?