Probability is favorable outcomes over total outcomes, and almost every ACT probability question is really a careful counting question wearing a disguise. If you can count what you want and count everything that could happen, you are done. The arithmetic afterward is usually a single fraction.
The definition, and what it guarantees
The probability of an event is the number of outcomes you want divided by the number of outcomes that are possible. Two useful consequences follow immediately. Every probability sits between 0 and 1, so an answer choice larger than 1 is wrong on sight. And a probability of 0 means impossible while 1 means certain, which gives you a quick sanity check on anything you compute.
Statistics and Probability is 12 to 15 percent of the Math section, and probability questions in that group tend to be short. They reward precision, not cleverness.
Start by counting the total
A bag holds 5 red marbles, 3 blue marbles, and 4 green marbles. The total is 5 + 3 + 4 = 12, and that 12 is the denominator for every question about a single draw from this bag.
- The probability of drawing blue is 3 out of 12, which simplifies to 1/4.
- The probability of drawing red is 5 out of 12, which does not simplify.
- The probability of drawing red or green is 9 out of 12, which simplifies to 3/4.
The trap is never the fractions. It is writing 3 out of 9 for blue because you forgot a group when you added.
The complement saves time
The probability that something does not happen is 1 minus the probability that it does. When a question asks for "at least one" of something, or lists four ways to succeed and one way to fail, compute the failure and subtract.
Back to the bag: the probability of not drawing blue is 1 minus 1/4, which is 3/4. Counting directly gives 9 non-blue marbles out of 12, which is the same 3/4, just with more steps.
And means multiply, or means add carefully
For independent events, where one result has no effect on the other, the probability that both happen is the product. Flip a fair coin and roll a standard six-sided die: the probability of heads and a 5 is 1/2 times 1/6, which is 1/12.
Repeated draws are independent only when you replace what you drew. Draw a marble from that bag, put it back, and draw again: the probability of red both times is 5/12 times 5/12, which is 25/144. Without replacement the second draw changes, so the probability of red both times is 5/12 times 4/11, which is 20/132, or 5/33.
For or, adding works only when the two events cannot both happen. Roll one die and ask for the probability of an even number or a number greater than 4. The even results are 2, 4, and 6. The results greater than 4 are 5 and 6. Adding 3/6 and 2/6 gives 5/6, but that double counts the 6. The actual set is 2, 4, 5, 6, so the answer is 4/6, or 2/3.
Probability from a table
A survey of 50 students reports whether they play an instrument. Among the 30 juniors, 18 play and 12 do not. Among the 20 seniors, 14 play and 6 do not. That means 32 students play in total and 18 do not.
- Probability a random student is a senior who plays: 14 out of 50, which is 7/25, or 28 percent.
- Probability a random student plays at all: 32 out of 50, which is 16/25, or 64 percent.
- Probability a student plays given that the student is a senior: now the pool is only the 20 seniors, so it is 14 out of 20, which is 7/10, or 70 percent.
Those last two look alike and are not. The word given, or a phrase like among the seniors, shrinks the denominator to a single row or column.
Common traps
- Miscounting the total. Add every category, including the one the question is about.
- Confusing and with or. Both events happening is a product. Either event happening is a sum, minus any overlap.
- Ignoring replacement. If the item is not put back, both the top and the bottom of the second fraction drop by one in the relevant category.
- Using the grand total for a conditional question.A question restricted to one group uses that group's total.
- Leaving it blank. There is no penalty for a wrong answer, so eliminate anything above 1 or obviously too large and pick from what remains.
A short drill
Using the bag of 5 red, 3 blue, and 4 green, answer four in a row: green, not green, green twice with replacement, and green twice without replacement. You should get 4/12 or 1/3, then 2/3, then 1/3 times 1/3 which is 1/9, then 4/12 times 3/11 which is 12/132, or 1/11.
Probability shares a category with mean, median, mode, and range, and questions often pull from both, so learn them together. The full map of that category is in statistics and probability on the ACT, and if your misses are counting slips rather than concept gaps, common ACT Math mistakes is worth a read. thirty-six quizzes come with worked explanations if you want a steady supply to practice on.
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