Quick answer
For a question that asks for the least and greatest probability that both Company X (50%) and Company Y (60%) post a profit:
Least probability for both: 10%.
Greatest probability for both: 50%.
The greatest overlap is the smaller of the two probabilities, min(50%, 60%) = 50%. The least overlap is max(0, 50% + 60% − 100%) = 10%. The rest of this guide shows exactly why, and why the other choices — 5%, 25%, 60%, and 80% — are traps.
The full question
According to a prominent investment adviser, Company X has a 50% chance of posting a profit in the coming year, whereas Company Y has a 60% chance of posting a profit in the coming year.
Select for Least probability for both the least probability, compatible with the probabilities provided by the investment adviser, that both Company X and Company Y will post a profit in the coming year. And select for Greatest probability for both the greatest probability, compatible with the probabilities provided by the investment adviser, that both Company X and Company Y will post a profit in the coming year. Make only two selections, one in each column.
Answer choices: 5% · 10% · 25% · 50% · 60% · 80%
This is a two-part (two-column) selection format: one answer for the smallest the joint probability could be, and one for the largest. You make exactly one selection in each column from the same shared list of choices.
Step-by-step solution
Step 1: Name the events and translate the prompt
Use clean notation. Let:
- A = the event that Company X posts a profit, with P(A) = 50%
- B = the event that Company Y posts a profit, with P(B) = 60%
The question asks for the possible values of P(A and B) — the probability that both companies profit. Crucially, the adviser never says the two outcomes are independent. So we are not allowed to multiply 50% × 60%. Instead we must ask: across every scenario consistent with these two numbers, how small and how large can the overlap be?
Step 2: Find the greatest possible overlap
The two events overlap as much as possible when every profitable outcome for the smaller event also counts as a profitable outcome for the larger one. Company X profits only 50% of the time. Even if all of that 50% sits inside Company Y's 60%, the "both profit" slice cannot grow past 50% — there simply is no more Company-X profit to put there.
Greatest P(A and B) = min(P(A), P(B)) = min(50%, 60%) = 50%.
You can never have both events happen more often than the rarer event happens on its own. That caps the overlap at 50%.
Step 3: Find the least possible overlap
Now push the two events as far apart as possible. If they never overlapped, together they would cover 50% + 60% = 110% of all outcomes. But a sample space only holds 100%. The excess — 110% − 100% = 10% — has nowhere to go except into the overlap. So the two events are forced to share at least 10%.
Least P(A and B) = max(0, P(A) + P(B) − 100%) = max(0, 50% + 60% − 100%) = max(0, 10%) = 10%.
Step 4: State the two selections
Least probability for both → 10%. Greatest probability for both → 50%.
So the joint probability that both companies post a profit is not a fixed number — it can be anything from 10% to 50%, depending on how correlated the two outcomes turn out to be.
The theory: least and greatest overlap in probability
This question is a clean test of how two events can sit inside one sample space. Picture a Venn diagram: a circle for A (area 50%) and a circle for B (area 60%) drawn inside a box whose total area is 100%.
Maximum overlap: slide the small circle inside the big one
Slide Company X's 50% circle completely inside Company Y's 60% circle. Now the overlap equals the entire smaller circle, 50%. You cannot do better: the overlap is part of circle A, so it can never be larger than A itself. Hence the ceiling is min(P(A), P(B)).
Minimum overlap: pull the circles apart until the box is full
Now drag the circles apart to minimize the shared region. They would love to be disjoint, but 50% + 60% = 110% of area cannot fit in a 100% box without stacking. The two circles must spill into each other by at least 10%. Hence the floor is max(0, P(A) + P(B) − 100%). (If the two probabilities had summed to 100% or less — say 30% and 40% — the floor would be 0%, because disjoint circles would fit with room to spare.)
Where the formulas come from
Both bounds fall straight out of the inclusion–exclusion rule:
P(A or B) = P(A) + P(B) − P(A and B).
Rearranging, P(A and B) = P(A) + P(B) − P(A or B). Since P(A or B) can be at most 100%, the overlap is at least P(A) + P(B) − 100% = 10%. Since P(A or B) is at least the larger single probability (60%), the overlap is at most P(A) + P(B) − 60% = 50%. Same two answers, derived algebraically.
Why the answer is not 0%, 25%, 60%, or 80%
| Tempting choice | Why a test-taker picks it | Why it is wrong |
|---|---|---|
| 0% for the least | Assume the two events could be mutually exclusive, so "both profit" never happens. | They cannot be disjoint. 50% + 60% = 110% > 100%, so at least 10% must overlap. The floor is 10%, not 0%. (0% is not even offered as a choice — a deliberate nudge.) |
| 5% for the least | Misremember the formula or halve the 10% excess. | The forced overlap is exactly the full excess over 100%: 110% − 100% = 10%. There is nothing that produces 5%. |
| 25% (= 50% × 60% rounded, or a "middle" guess) | Treat the events as independent and multiply, getting 30%, then round, or just split the difference. | The problem never says the outcomes are independent, so you may not multiply. 25% (the independent value is actually 30%) is one possible point inside the range, not the least or the greatest. |
| 60% for the greatest | Grab Company Y's larger probability as the ceiling. | The overlap is part of Company X's 50% region too, so it cannot exceed 50%. The cap is the smaller probability, not the larger one. |
| 80% for the greatest | Add or otherwise combine the two percentages. | A joint "both" probability can never exceed either individual probability, let alone their sum. 80% is impossible here. |
GMAT Focus Quant strategy for overlap and probability questions
- Check for an independence claim before you multiply. P(A and B) = P(A) × P(B) is legal only when the problem states or implies independence. "Least/greatest compatible with" language is a flag that you are being asked for a range instead.
- Cap the overlap at the smaller probability. The greatest possible P(A and B) is always min(P(A), P(B)). The "both" slice lives inside each circle, so it can't beat the smaller one.
- Floor the overlap with the excess over 100%. The least possible P(A and B) is max(0, P(A) + P(B) − 100%). If the two add to 100% or less, the floor is 0%.
- Draw the Venn box. Two circles in a 100% box make both extremes obvious: slide together for the max, pull apart for the min.
- Match each answer to its column. In two-part selection questions, confirm you put the smaller value under "least" and the larger under "greatest" — transposing them is the most common careless error.
Want a tutor to drill probability and overlap reasoning until the "min for the max, excess for the min" move is automatic? MBA House runs live GMAT Focus prep and private tutoring in New York built around clean reasoning, not memorized tricks.
How MBA House NYC teaches this type of GMAT question
Probability questions like this one reward the habits MBA House teaches in every GMAT Focus class: translate the words into events and probabilities, ask whether the problem actually gave you independence, and reach for a picture — here, two circles in a 100% box — before reaching for a formula. Our New York students learn to recognize "least/greatest compatible with" as a signal that the answer is a range bounded by min(P(A), P(B)) on top and max(0, P(A) + P(B) − 1) on the bottom, so a question that looks open-ended collapses into two quick calculations.
If you want the full picture of the current exam, start with what the GMAT is and our breakdown of the GMAT Focus Edition. For more worked Quant, see our mixture profit Data Sufficiency question, our profit question with tiered costs, and our percent error Data Sufficiency question, plus our sets from the Official Guide Quantitative Review 2026–2027 — including all 25 questions explained and the 17 easy Official Guide Quant questions. To turn practice into a score, our GMAT Focus tutor NYC page explains how live classes and private tutoring work, and our guide to building GMAT and admissions strategy together ties your target score to your school list.
For the official source on the exam and its Quant section, see the GMAC overview at mba.com/exams/gmat-exam and the Quantitative Reasoning prep page.
Preparing for the GMAT in New York? MBA House offers personalized GMAT Focus tutoring with proven score-improvement strategies and weekly live Quant practice.
GMAT probability overlap FAQs
What is the answer to the company profit probability question?
The least possible probability that both companies post a profit is 10%, and the greatest is 50%. With P(X) = 50% and P(Y) = 60%, the greatest overlap is min(50%, 60%) = 50% and the least overlap is max(0, 50% + 60% − 100%) = 10%.
Why is the greatest probability 50% and not 60%?
The overlap can never exceed the smaller of the two probabilities. Company X profits only 50% of the time, so even if every X-profit outcome also lands inside Company Y's 60%, the "both profit" probability cannot rise above 50%.
Why is the least probability 10% and not 0%?
The two probabilities add to 110%, more than a 100% sample space can hold without overlapping. The excess, 110% − 100% = 10%, is forced into the overlap, so the two events must share at least 10%.
What formula gives the minimum and maximum overlap?
For events A and B, the maximum joint probability is min(P(A), P(B)), and the minimum is max(0, P(A) + P(B) − 1). Both follow from inclusion–exclusion, P(A or B) = P(A) + P(B) − P(A and B), and the fact that no probability exceeds 1.
Is this a hard GMAT Focus probability question?
It is medium difficulty. The arithmetic is trivial, but it tests whether you know that without independence the joint probability is a range, and whether you can find that range's extremes with Venn diagram logic — exactly the reasoning GMAT Focus Quant rewards.