Probability When the Sample Space Is Not Given

Most probability questions you can solve in one step are secretly two questions. The real work is figuring out what the full set of equally likely outcomes even is. Once you have that, dividing favorable outcomes by total outcomes is the easy part. The trap is rushing to that division before you've pinned down the sample space correctly, which means you count the wrong total, the wrong favorable cases, or both.

Build the sample space by hand when none is given. For two dice, list ordered pairs: (1,1), (1,2), (1,3), and so on through (6,6), giving 36 total outcomes, since each die has 6 faces and the draws are independent. A tree diagram works the same way for multi-stage situations: each branch is a choice, and following every path to its end gives you every outcome. For something like coin flips or drawing cards in sequence, write out the stages one at a time rather than trying to picture the whole thing at once.

Drawing without replacement changes the sample space between stages, and this is where the topic connects to conditional probability and independence. If a bag has red and blue marbles and you draw one without putting it back, the second draw happens from a smaller bag: one fewer marble overall, and one fewer of whichever color came out first. That means the two draws are not independent, and you cannot just multiply the same fraction twice. This idea of a shrinking, changing sample space also sets up expected value and more advanced counting, where keeping track of exactly what's left at each stage is the whole game.