Understanding the Probability of Scoring Zero in a Guessing Quiz

A multiple choice trivia quiz has ten questions, each with four possible answers. If someone simply guesses at each answer, what is the probability of then getting a zero? This article delves into the calculations and provides a clear understanding of the probability involved.

Probability Calculation

The probability of guessing k correct answers out of ten is given by the formula:

P(k) 10!/(k!(10-k)!).k.0.25k.0.7510-k
Forgetting the factorials, we can simplify our calculation for scoring zero:

To find the probability of getting zero correct, we need to calculate the chances of all answers being wrong. The probability of each question being wrong is 75%, or 0.75. Thus, the probability of getting all ten questions wrong is:

0.7510
This can also be expressed as a fraction: (0.75)10 59049/1048576
Or as a decimal: 0.0563, approximately 5.63%

Real-World Application and Context

If you give the test to 18 different people who will be guessing, the odds are that one of them will get a zero. Considering the probability of scoring zero as 5.63%, out of 18 guesses, we expect one person to get all questions wrong. This is based on the binomial distribution principle.

Further Extension: The Binomial Distribution

Let’s consider the scenario of a student who is guessing the answers. The probability of a student getting a single question wrong is 75%, or 0.75. If we want to find the probability of getting none right, we need to multiply 0.75 ten times. This can be written as:

(0.75)10
Which is approximately 0.0563, or about 5.63%.

Alternatively, the probability of a student getting no questions correct by guessing can be calculated using the P(x 0) formula in the binomial distribution, where there are 10 trials and the probability of success (getting an answer correct) is 1/4 (or 0.25).

Calculation in R

We can use the R programming language to calculate the same probabilities. Below is the R code and resulting output:

R Core Team (2023). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
dbinom(0, 10, 0.25)
[1] 0.05631351
This is the same as:
0.75^10
[1] 0.05631351

Conclusion

When guessing on a multiple choice quiz with 10 questions and 4 possible answers for each, the probability of scoring a zero is approximately 5.63%. While the chance of getting a perfect score is exceedingly low (1 in 1,048,576), guessing can still result in no correct answers due to the large number of questions and the random distribution of correct and incorrect responses.

Understanding these probabilities is crucial for students and educators alike, providing a practical application of basic statistical principles in everyday scenarios.