Probability without replacement
A single draw is favorable outcomes over total outcomes. The questions that give students trouble are the ones with two draws, or a condition, or the phrase at least one, and each of those has one specific adjustment. Learn the three adjustments and the topic closes.
One draw, one fraction
Count the outcomes you want, count all the outcomes, and divide. Three red beads among ten is three tenths. The only mistake available here is counting the wrong total, which is why reading what the group is matters more than the arithmetic. If the question says from the red beads, the total is the red beads, not the whole jar.
Two draws: check whether the first item goes back
With replacement, the second draw starts from the same collection, so the two fractions are identical and independent. Without replacement, the first item stays out, so both the number you want and the total drop by one. A bag with 5 red and 3 blue, drawing two without replacement, gives a first red of 5/8 and a second red of 4/7, and the probability of both is their product, 20/56, which simplifies to 5/14.
A condition changes the denominator
The phrase given that, or a question that names a subgroup, resets the denominator. Of 100 students, suppose 40 play an instrument and 24 of those also read weekly. The probability that a student reads, given that they play an instrument, is 24/40, not 24/100. Two-way tables exist on this test to make that substitution visible, so locate the row or column named in the condition and use its total.
- Without a condition: divide by the grand total
- Given a row: divide by that row's total
- Given a column: divide by that column's total
At least one: solve the opposite
Probability of at least one is one minus the probability of none. Two rolls of a die landing at least one six is one minus five sixths squared, which is eleven thirty sixths. The direct count is possible and slower, and it is easy to double count, which is why the complement is the method to reach for whenever the phrase at least appears.
Independent or not
Two events are independent when one happening does not change the other's probability. Multiplication is only valid for the joint probability when that is true, or when the second probability is already conditional: the probability of A and B is the probability of A times the probability of B given A. That single sentence covers replacement, no replacement, and independent events in one rule.
What changes when there is no replacement?
The second draw has one fewer item in total and one fewer of the item already taken. Both the numerator and the denominator drop by one for the second fraction.
How do I read a two-way table for probability?
Find the group named in the question and use its total as the denominator. The cell at the intersection of the two named categories is the numerator.
What does given that mean in a probability question?
It narrows the group. You are no longer picking from everyone, only from the named subgroup, so that subgroup becomes the denominator.
Why is at least one easier with the complement?
Because the opposite of at least one is none, and none is a single calculation. One minus the probability of none gives the answer without counting every possible combination.
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