Why Guessing a Multiple‑Choice Answer with Two Coin Tosses Isn’t 1⁄16

During a teacher‑student discussion, a student asked whether the probability of correctly guessing a multiple‑choice answer (four options) by tossing two coins—producing four possible coin outcomes—was 1/16, calculated as (1/4)*(1/4) under the assumption that the coin toss and the answer are independent.

Four teachers offered different explanations:

Teacher 1: The coin toss and the answer are merely two ways of describing the same event, so the probability remains 1/4.

Teacher 2: The coin toss is a random event with probability 1/4 for each option, while the correct answer is predetermined (probability 1). Multiplying independent events gives 1/4; the student’s calculation is wrong.

Teacher 3: This is a conditional probability problem; given that the correct answer is A, the probability of guessing A is 1/4.

Teacher 4: To avoid confusion, the probability space must be explicitly written. One basic event space is the set of all ordered pairs (student guess, correct answer): {(A,A),(B,A),(C,A),(D,A),(A,B),(B,B),(C,B),(D,B),(A,C),(B,C),(C,C),(D,C),(A,D),(B,D),(C,D),(D,D)}. Another, smaller space assumes the correct answer is fixed: {(A,A),(B,A),(C,A),(D,A)}. In the first space, the chance of the student guessing correctly is 1/16; in the second, it is 1/4.

If the student then asks: when the coin toss yields A but the correct answer is B, what is the probability of a correct guess? The answer is clearly 0.

The discussion shows that different interpretations arise from different probability spaces. Teacher 4’s approach, listing all basic events, provides the most fundamental and compatible framework for understanding concepts such as independent events, conditional probability, and sample spaces.

By constructing the appropriate sample space, educators can clarify assumptions, premises, and conditions, helping students grasp why the correct probability is 1/4 rather than 1/16 when the answer is fixed.