TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW03.1

Multiple-choice exams

Nik takes an exam in Statitstics and Probability Theory that contains 20 multiple-choice questions. Each question has four possible options and only one of the four options is correct. He knows the answer to ten questions, but he has no idea about the other ten questions so he chooses answers randomly. His score $X$ on the exam is the total number of answers he answered correctly.

(a) Find the probability mass function of $X$.

(b) Compute the probability $P(X>16)$.

Dieses Beispiel ist als solved markiert. Ist dies falsch oder ungenau? Aktualisiere den Lösungsstatus (Details: Vorlage:Beispiel)

Lösungsvorschlag von Lessi

--Lessi 2024-02-07T13:04:11Z

a)

$X$ is the number of correct answers on the exam, with 10 being certainly correct and 10 chosen randomly with a probability of guessing correct being $p={\frac {1}{4}}$. Therefore $X\sim 10+B(10,{\frac {1}{4}})$. This means for the pm $p(x)$ that we need to shift $x-10$ to accommodate for getting less than 10 correct answers being an impossible event:

$p(x)=P(X=x):={\binom {10}{x-10}}p^{x-10}(1-p)^{20-x}$

val <- 0:20

p <- function(x) {
  dbinom(x - 10, 10, 1/4)
}

plot(p(val), type='h', xlab='correct questions', ylab='probability')

b)

F <- function(x) {
  pbinom(x - 10, 10, 1/4)
}

plot(F(val), type='s', xlab='correct questions', ylab='probability')

# Probability of getting more than 16 correct answers
1 - F(15)
1 - pbinom(5, 10, 1/4) # checking with a "unshifted" distribution