Question

hw 9 - binomial distributions
score: 2.78/10 answered: 3/10
question 6
suppose you are taking an exam with 10 questions. each question has 4 possible choices. if you are just guessing at the answers,
a) what is the probability you get exactly 2 answers correct? ( p(x = 2) ) (round to 4 decimals)
b) what is the probability you get at most 2 answers correct? ( p(x leq 2) ) (round to 4 decimals)
question help: video message instructor

Explanation:

Part A

Step1: Identify binomial parameters

We have a binomial distribution with \( n = 10 \) (number of questions), \( p=\frac{1}{4}=0.25 \) (probability of guessing a question correctly), and we want \( P(X = 2) \). The binomial probability formula is \( P(X=k)=\binom{n}{k}p^{k}(1 - p)^{n - k} \), where \( \binom{n}{k}=\frac{n!}{k!(n - k)!} \).

Step2: Calculate combination

First, calculate \( \binom{10}{2}=\frac{10!}{2!(10 - 2)!}=\frac{10\times9}{2\times1}=45 \).

Step3: Calculate probability terms

Then, \( p^{k}=(0.25)^{2}=0.0625 \) and \( (1 - p)^{n - k}=(0.75)^{8}\approx0.100113 \).

Step4: Multiply to find \( P(X = 2) \)

Multiply these together: \( P(X = 2)=45\times0.0625\times0.100113\approx0.2816 \).

Step1: Recall at most 2 means \( X = 0,1,2 \)

We need to calculate \( P(X\leq2)=P(X = 0)+P(X = 1)+P(X = 2) \).

Step2: Calculate \( P(X = 0) \)

Using the binomial formula, \( \binom{10}{0}=1 \), \( (0.25)^{0}=1 \), \( (0.75)^{10}\approx0.056314 \), so \( P(X = 0)=1\times1\times0.056314 = 0.056314 \).

Step3: Calculate \( P(X = 1) \)

\( \binom{10}{1}=\frac{10!}{1!(10 - 1)!}=10 \), \( (0.25)^{1}=0.25 \), \( (0.75)^{9}\approx0.075085 \), so \( P(X = 1)=10\times0.25\times0.075085\approx0.187713 \).

Step4: Add the probabilities

We already know \( P(X = 2)\approx0.2816 \) from part A. Now sum them: \( 0.056314+0.187713 + 0.2816\approx0.5256 \).

Answer:

\( 0.2816 \)

Part B