Probability of Two Independent Events on GMAT

50-Word Summary

Independent events are ones where one happening does not change the probability of the other. P(A and B) = P(A) × P(B); P(neither) = [1 − P(A)] × [1 − P(B)]; P(at least one) = 1 − P(neither); P(at most one) = 1 − P(A)P(B).

Overview

Independent events are events where knowing that one happens does not change the chance of the other. In GMAT probability, think in simple blocks. For the chance that both happen, consider each event separately and then multiply their chances. For the chance that neither happens, multiply their failure chances; use 1 minus that to find “at least one.” The same idea gives “exactly one” without long case lists. Before computing, check whether independence is stated; if it is not stated, do not assume it. In your GMAT prep course, practice turning short word statements into clean expressions. Later, confirm timing and accuracy with GMAT mock tests; always analyze your performance carefully and learn from mistakes.

 

The Concept of Independent Events

Two events are independent if knowing that one happens does not change the probability of the other.

Formally:

• P(A | B) = P(A) and P(B | A) = P(B)

P(A | B) = P(A) means that even after B happens, the chance of A stays the same.

P(B | A) = P(B) says the same in reverse.

When both hold, A and B are independent; they do not affect each other.

Example: a coin flip and a die roll.

• P(A and B) = P(A) × P(B)

Means: if events are independent, the chance that both happen equals the product of their separate chances.

Example:

Coin shows Heads (1/2) and die shows 3 (1/6).

Both together: 1/2 × 1/6 = 1/12.

(Multiply because one does not affect the other outcome).

Probability Scenarios for Two Independent Events

Assume A and B are independent with probabilities P(A) and P(B).

• Both happen (A and B): P = P(A) × P(B)
• A happens, B doesn’t: P = P(A) × [1 − P(B)]
• B happens, A doesn’t: P = [1 − P(A)] × P(B)
• Neither happens: P = [1 − P(A)] × [1 − P(B)]
• Exactly one happens: P = P(A) × [1 − P(B)] + [1 − P(A)] × P(B)
• At least one happens (A or B): P = 1 − [1 − P(A)] × [1 − P(B)]
• At most one happens: P = 1 − P(A) × P(B)

An Explanatory Example of Probability of Two Independent Events

Question: Probability that A will shoot a target is 1/3. Probability that B will shoot the same target is 1/2. What is the probability that…

1. A hits, B doesn’t.
2. Both hit.
3. None hits.
4. The target is hit.
5. Exactly one hits.

Probability is the measure of how likely an event is to occur. In every case, it is expressed as the ratio of favorable outcomes to total possible outcomes. In this example, A has a probability of 1/3 to hit the target, while B has a probability of 1/2.

Case 1: A hits, B does not

This is calculated as A × (1 − B)

= (1/3) × (1/2)

= 1/6.

Case 2: Both hit

The probability is A × B

= (1/3) × (1/2)

= 1/6.

Case 3: Neither hits

The complement values are (1 − A) × (1 − B)

= (2/3) × (1/2)

= 1/3.

Case 4: The target is hit (at least once)

Instead of summing three cases, it is easier to take 1 − P(neither hits).

That is 1 − 1/3

= 2/3.

Case 5: Exactly one hits

This is A × (1 − B) + (1 − A) × B

= (1/3 × 1/2) + (2/3 × 1/2)

= 1/2.

Key Takeaways

Independent events do not affect each other. Build problems with a clean setup. For the chance that both happen, multiply their chances. For neither, multiply their failure chances. Use one minus neither for at least one. For exactly one, add A and not B with not A and B. At most one excludes both. Check whether independence is given; do not assume it. Write probabilities in symbols and words so steps remain traceable. Build habits by solving a small set daily, then rechecking errors. Use GMAT simulation to rehearse under timed conditions and to confirm that the expressions feel natural.

Parting Thought

Independent events teach a calm lesson: focus on what you can control, treat pieces clearly, then combine them with care. The same habit strengthens GMAT preparation: learn one idea well, verify it, and add the next without clutter. It also steadies the MBA admissions process: scores, essays, recommendations, and interviews improve when each part receives honest attention. Small, independent wins multiply into strong outcomes. When a result feels uncertain, use the complement idea in spirit: reduce what can go wrong, and what can go right grows. Clarity, patience, and steady practice turn probability rules into a way of working and a way of living.