Question

The probability that you guess correctly on any given question is 20% (since there are five choices on each question). What is the probability that you are able to guess ten or more correct answers?

Answer 1

The probability that you are able to guess ten or more correct answers is P(x≥10) = 0.0026

Explanation:

The question is incomplete:

You take a multiple choice test that you are not prepared for, so you have to guess on all twenty questions. The probability that you guess correctly on any given question is 20% (since there are five choices on each question). What is the probability that you are able to guess ten or more correct answers?

This can be modeled by a binomial random variable, with sample size n=20 and probabillity of success p=0.2.

The probability of guessing k answers right can be calculated as:

`P(x=k)=\dbinom{n}{k}p^k(1-p)^(n-k)=\dbinom{20}{k}\cdot0.2^k\cdot0.8^(20-k)`

We have to calculate the probabiltiy that 10 or more answers are correctly answered guessing: P(x≥10)

`P(x\geq10)=\sum_(k=10)^(20)P(x=k)`

`P(x=10)=\dbinom{20}{10}\cdot0.2^(10)\cdot0.8^(10)=184756\cdot0.0000001\cdot0.1074=0.0020\\\\\\P(x=11)=\dbinom{20}{11}\cdot0.2^(11)\cdot0.8^(9)=167960\cdot0.00000002\cdot0.1342=0.0005\\\\\\P(x=12)=\dbinom{20}{12}\cdot0.2^(12)\cdot0.8^(8)=125970\cdot0\cdot0.1678=0.0001\\\\\\P(x=13)=\dbinom{20}{13}\cdot0.2^(13)\cdot0.8^(7)=77520\cdot0.000000001\cdot0.2097=0.0000\\\\\\P(x\geq14)=0.0000`

`P(x\geq10)=0.0020+0.0005+0.0001=0.0026`