Three unbiased coins are tossed simultaneously. What is the probability of getting at most two heads?

Introduction / Context:
This question focuses on probability for repeated coin tosses. The phrase at most two heads means any outcome where the number of heads is zero, one or two, but not three. Because the coins are unbiased, every possible sequence of heads and tails is equally likely, and we can use either direct counting or the complement rule to find the required probability.

Given Data / Assumptions:

• There are three fair coins.
• Each coin has two possible outcomes: head or tail.
• All 2^3 = 8 outcomes are equally likely.
• Event of interest: number of heads is less than or equal to two.

Concept / Approach:
The easiest way is to use the complement rule. The only way to have more than two heads with three coins is to have exactly three heads. So at most two heads is the complement of exactly three heads. We find the probability of three heads and subtract it from one. This saves time compared to listing all the zero, one and two head cases separately, although we can list them later as a check.

Step-by-Step Solution:
Total number of outcomes = 2^3 = 8. The only outcome with three heads is HHH, so there is exactly 1 outcome with three heads. Probability of exactly three heads = 1 / 8. Probability of at most two heads = 1 - probability of three heads. So probability of at most two heads = 1 - 1 / 8 = 7 / 8.

Verification / Alternative check:
We can list all outcomes explicitly: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. Outcomes with at most two heads are HHT, HTH, THH, HTT, THT, TTH, TTT, which is 7 outcomes. Dividing by the total 8 outcomes gives 7 / 8, which matches the complement method and confirms the result.

Why Other Options Are Wrong:
3/4 would correspond to only 6 favourable outcomes out of 8, but we have 7. 2/3 would require about 5.33 favourable outcomes, which does not make sense for a discrete sample of 8 equally likely cases. 5/8 would mean only 5 favourable outcomes, again not matching the correct count.

Common Pitfalls:
Learners sometimes misread at most two heads as exactly two heads. That would give only 3 outcomes and a probability of 3 / 8, which is incorrect. Others forget that three heads is only a single outcome and assume it has a larger probability. Carefully distinguishing between complement events and counting all possibilities avoids these errors.

Final Answer:
The probability of getting at most two heads when three fair coins are tossed is 7/8.