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)!k!}
\end{aligned}
[/tex]
Answer (1)
Use the binomial probability formula: P ( k successes ) = n C k p k ( 1 − p ) n − k .
Identify n = 8 , k = 1 , and p = 4 1 .
Calculate P ( 1 success ) = 8 C 1 ( 4 1 ) 1 ( 4 3 ) 7 .
The probability is approximately 0.267 .
Explanation
Understand the problem We are given a problem where Van guesses on all 8 questions of a multiple-choice quiz. Each question has 4 answer choices. We need to find the probability that he gets exactly 1 question correct. We can use the binomial probability formula to solve this problem.
Recall the binomial probability formula The binomial probability formula is given by: P ( k successes ) = n C k p k ( 1 − p ) n − k where:
n is the number of trials (questions).
k is the number of successes (correct answers).
p is the probability of success on a single trial (probability of guessing correctly).
n C k is the number of combinations of n items taken k at a time.
Identify the values for n, k, p, and 1-p In this problem:
n = 8 (number of questions)
k = 1 (number of correct answers)
p = 4 1 = 0.25 (probability of guessing correctly on a single question)
1 − p = 4 3 = 0.75 (probability of guessing incorrectly on a single question)
Calculate the number of combinations First, we calculate the number of combinations: n C k = 8 C 1 = ( 8 − 1 )! 1 ! 8 ! = 7 ! 1 ! 8 ! = 7 ! × 1 8 × 7 ! = 8 So, 8 C 1 = 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 ≈ 0.13348388671
Calculate the final probability Now, we plug these values into the binomial probability formula: P ( 1 success ) = 8 × 4 1 × ( 4 3 ) 7 = 8 × 0.25 × 0.13348388671 P ( 1 success ) = 2 × 0.13348388671 = 0.26696777342 Rounding to the nearest thousandth, we get 0.267 .
State the final answer Therefore, the probability that Van gets exactly 1 question correct is approximately 0.267.
Examples
Consider a quality control scenario where you inspect 8 items from a production line, and each item has a 25% chance of being defective. The probability of finding exactly 1 defective item among the 8 is analogous to Van's quiz problem. This helps assess the reliability of the production process.
