Question

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? You must show correct inputs to either binompdf or binomcdf to receive credit.

Answer 1

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

Explanation:

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

The probability of getting 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)`

Now, we have to calculate the probabiltiy that 10 or more answers are correctly answered guessing. This is P(x≥10).

`P(x\geq10)=P(x=10)+P(x=11)+P(x=12)+P(x=13)+P(x\geq14)`

Note: the expression is simplified for x≥14 because we know the additional probability is less than 0.00005.

`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+0.0000+0.0000=0.0026`

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