Casework in Probability

Some events can happen in more than one mutually exclusive way, such as a dice sum of 7 or 11. This lesson splits such events into disjoint cases, finds each case's probability separately, and adds them together using the addition rule for events that cannot both occur.

What you will learn

• Split a probability problem into disjoint cases
• Compute each case's probability separately
• Add case probabilities to find the total

Keep paper and a pencil nearby. Try each problem before opening a hint, and mark sections reviewed as you work through them.

Recognizing when a problem needs cases

An event needs casework when it can occur through more than one distinguishable route, and no single outcome can satisfy more than one of those routes at once. For example, "the sum of two dice is 7 or 11" is really two separate events — sum equals 7, and sum equals 11 — and a single roll can produce only one sum, so the two routes never overlap.

The signal to look for is the word "or" connecting two conditions that describe different, non-overlapping outcomes. If the two conditions could both be true at the same time for some outcome, the event is not a clean case split and needs a different technique to avoid double-counting.

Computing each case's probability separately

Once the cases are identified, each one is treated as its own probability problem: build or reuse the sample space, count the favorable outcomes for that specific case, and divide by the total. The methods from the previous two lessons — listing an explicit sample space and labeling identical-looking objects individually — apply within each case exactly as before.

It helps to keep the total sample space size the same across all cases being compared, since the cases will eventually be combined over that same total. For two dice, every case uses the same 36-outcome sample space; only the favorable count changes from case to case.

Adding case probabilities: the addition rule for disjoint events

Because the cases cannot overlap, the total probability of the combined event is simply the sum of the individual case probabilities: $P(\text{case 1 or case 2})=P(\text{case 1})+P(\text{case 2})$. This works because adding disjoint favorable counts and then dividing by the shared total gives the same result as dividing each case first and then adding.

This addition rule fails if the cases are not actually disjoint, since an outcome belonging to both cases would then be counted twice. Before adding, it is worth double-checking that no single outcome could satisfy more than one case, the same exhaustive, non-overlapping standard used for casework in counting problems.

See the reasoning unfold

Pause after each step and predict what comes next.

Common mistakes to catch

• Splitting into cases that actually overlap, which double-counts some outcomes when the probabilities are added.
• Missing a case entirely, which undercounts the true probability.
• Forgetting to recompute the total sample space size for a combined experiment, such as a coin and a die together.

Try with support

Guided practice

Choose a strategy, take a first step, and use the hints when you need direction.

Put it into practice

Independent practice

Work on your own first. If you reveal a solution, work through it and mark it reviewed.

Bring it together

• Casework in probability splits an event into mutually exclusive routes and adds each route's probability.
• Every case must be checked to be truly disjoint from every other case, or the sum overcounts.
• The same casework idea applies to single experiments, combined experiments, and multi-object draws alike.