In a multiple choice quiz each question has 5 alternatives out of them only one answer is correct. What is the probability of 6 correct answers out of 10 questions?

Question

Accepted Answer

The correct option is 3:
This problem is a classic example of a **Binomial Probability Distribution** question. The conditions for a binomial distribution are met: there is a fixed number of trials ($n=10$ questions), each trial is independent, there are only two outcomes (correct or incorrect), and the probability of success is constant. Since there are 5 alternatives and only one is correct, the probability of a random guess being correct is $p = 1/5 = 0.2$. The formula for the binomial probability of exactly $k$ successes in $n$ trials is $P(X=k) = C(n, k) \* p^k \* (1-p)^{n-k}$. For 6 correct answers out of 10 questions ($n=10, k=6, p=0.2$): $P(X=6) = C(10, 6) \* (0.2)^6 \* (0.8)^4$. Calculating the combinations $C(10, 6) = 210$. The resulting probability is $210 \* 0.000064 \* 0.4096 = 0.005505024$. Rounded to four decimal places, the probability is $\mathbf{0.0055}$.

The incorrect option is 1 (0.55):
This option suggests an unrealistically high probability. A probability value must always be between 0 and 1, inclusive. The value 0.55 is plausible as a probability, but it is not the correct calculation for this scenario. If the probability of getting *one* question correct was 0.55, the total probability of getting 6 correct would be much lower than the simple average, since the formula involves multiplication of small probability terms raised to a power. The probability of 6 correct out of 10 for a single event chance of $0.2$ is very low, making 0.55 incorrect.

The incorrect option is 2 (0.055):
While this probability is much closer to the true value than Option 1, it is still incorrect. It suggests the probability is roughly ten times higher than the actual result. The calculation error leading to 0.055 might arise from incorrectly calculating the combinatorial term $C(10, 6)$ or making a mistake with the powers of $p$ and $(1-p)$. It is essential to ensure precise calculation of all components of the binomial probability formula, including the combinations and the probability components, to avoid this type of small but significant error.

The incorrect option is 4 (5.5):
This option is mathematically impossible as a probability. By definition, probability is a measure of the likelihood of an event occurring and must be expressed as a number between 0 (impossible event) and 1 (certain event). Any value greater than 1, such as 5.5, fundamentally violates the axioms of probability theory and cannot be the correct answer to a probability question. The value might be a result of a gross arithmetic error or a misinterpretation of the input values as percentages or raw counts, but it is not a valid probability.