A reliable method
Decide whether a draw is replaced before multiplying probabilities. For “at least one,” a complement often avoids listing several overlapping cases.
The common trap
Without replacement, both the favorable count and the total may change. A condition such as “given that it is blue” changes the denominator.
Make the connection
Write the event in words before computing. A probability must lie between zero and one; a count must be a nonnegative integer.
Worked example: choose the requested quantity
A fair spinner has 4 equal sectors, of which 2 are green. What is the probability of not landing on green? The green probability is its number of sectors divided by the total. Subtract that probability from 1 to obtain the complement. The requested value is 1/2.
Contrast and diagnose an error
A bag contains 3 blue counters and 4 red counters. Two counters are drawn with replacement. What is the probability both are blue? Replacement restores the original counts for the second draw. Multiply the same blue probability for the two independent draws. The requested value is 9/49. A tempting answer is 16/49: Compute two red draws instead.
Transfer practice and check
A box contains 4 marked cards and 3 unmarked cards. Two are drawn without replacement. What is the probability both are marked? Try it before reading on. Check: 2/7. For the first draw, use marked count over total count. After a marked first draw, reduce both the marked count and the total by one.
A fair spinner has 4 equal sectors, of which 2 are green. What is the probability of not landing on green?
- The green probability is its number of sectors divided by the total.
- Subtract that probability from 1 to obtain the complement.
- The requested value is 1/2.
Compare the other answer choices
Add the green count again in the denominator.
Assume a single nongreen sector.
Remove only one green sector.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
ACT Mathematics content