Van guessed on all 8 questions of a multiple-choice quiz. Each question has 4 answer choices. What is the probability that he got exactly 1 question correct? Round the answer to the nearest thousandth.
[tex]
\begin{aligned}
P(k \text { successes }) & = {}_n C_k p^k(1-p)^{n-k} \
{ }_n C_k & =\frac{n!}{(n-k)!\cdot k!}
\end{aligned}
[/tex]
Answer (2)
Define n = 8 , k = 1 , and p = 4 1 .
Calculate the binomial coefficient: 8 { C } 1 = 8 .
Calculate P ( 1 success ) = 8 × ( 4 1 ) × ( 4 3 ) 7 .
The probability that Van got exactly 1 question correct is 0.267 .
Explanation
Understand the problem We are given a binomial probability problem where Van guessed on all 8 questions of a multiple-choice quiz. Each question has 4 answer choices. We want to find the probability that he got exactly 1 question correct.
Define the variables Let's define the variables: n = 8 (number of trials, i.e., questions) k = 1 (number of successes, i.e., correct answers) p = 4 1 = 0.25 (probability of success on a single trial, i.e., guessing correctly) 1 − p = 4 3 = 0.75 (probability of failure on a single trial, i.e., guessing incorrectly)
State the binomial probability formula The binomial probability formula is given by: P ( k \t s u ccesses ) = n { C } k p k ( 1 − p ) n − k where n { C } k = ( n − k )! k ! n !
Calculate the binomial coefficient First, we calculate the binomial coefficient: 8 { C } 1 = ( 8 − 1 )! 1 ! 8 ! = 7 ! 1 ! 8 ! = 7 ! × 1 8 × 7 ! = 8
Calculate p^k Next, we calculate p k :
p k = ( 4 1 ) 1 = 4 1 = 0.25
Calculate (1-p)^(n-k) Then, we calculate ( 1 − p ) n − k :
( 1 − p ) n − k = ( 4 3 ) 8 − 1 = ( 4 3 ) 7 = ( 0.75 ) 7 Now, we calculate ( 0.75 ) 7 :
( 0.75 ) 7 ≈ 0.1334838867
Calculate the final probability Now, we plug these values into the binomial probability formula: P ( 1 \t s u ccess ) = 8 × ( 4 1 ) × ( 4 3 ) 7 = 8 × 0.25 × 0.1334838867 ≈ 0.2669677734 Rounding to the nearest thousandth, we get 0.267.
State the final answer Therefore, the probability that Van got exactly 1 question correct is approximately 0.267.
Examples
Consider a quality control process where a machine produces items, and each item has a 25% chance of being defective. If we inspect 8 items, the probability of finding exactly 1 defective item can be calculated using the same binomial probability formula. This helps in assessing the machine's performance and making necessary adjustments.
The probability that Van got exactly 1 question correct on a multiple-choice quiz with 8 questions, where each has 4 choices, is approximately 0.267. This is calculated using the binomial probability formula. We define the parameters, compute necessary components, and arrive at the final result through systematic calculations.
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