A condition changes the group from which you select. Count within that group before calculating a probability.
Conditional probability asks about an event within a specified group. “Given that” changes the denominator. The numerator counts members of that group who also meet the requested condition.
Worked example: read a two-way table
This table and all examples below are original teaching exercises, not released SAT questions.
| Art club | Not in art club | Total | |
|---|---|---|---|
| Music club | 12 | 18 | 30 |
| Not in music club | 8 | 22 | 30 |
| Total | 20 | 40 | 60 |
If one student is selected at random from the 60, the probability of belonging to both clubs is 12/60 = 1/5. Given that the student belongs to music club, the probability of also belonging to art club is 12/30 = 2/5. Given art club membership instead, the probability of music club membership is 12/20 = 3/5.
The overlap count is the same, but the conditioning group changes. These two conditional probabilities need not be equal.
Complements and “or”
Given music club membership, the probability of not belonging to art club is 18/30 = 3/5, also 1 − 2/5. For selection from all 60, the probability of belonging to at least one of the clubs is (30 + 20 − 12)/60 = 19/30. Subtract the overlap once because adding both totals counted it twice.
Independence needs evidence
The unconditional probability of art membership is 20/60 = 1/3, while its probability given music membership is 2/5. In this table, the two memberships are not independent. Do not multiply unconditional probabilities for an intersection unless independence is given or established.
Independence is different from mutually exclusive events. The two club memberships overlap, so they are not mutually exclusive. A student can meet both conditions.
Worked example: repeated selections
A bag contains 3 red and 2 blue counters. Without replacement, the probability of drawing two red counters is (3/5)(2/4) = 3/10. With replacement and a fresh random draw, it is (3/5)(3/5) = 9/25. Replacement determines whether the second group's composition changes.
A denominator check
Translate the question into “Among the ___, how many ___?” before calculating. Mark the condition in a table and identify whether the question concerns one selection, repeated selections, a complement, or a union.
For practice, log wrong denominator, reversed condition, double-counted overlap, and assumed independence. For proportions and units outside probability, use the ratios and rates guide.
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