An algebra 2 test has 5 multiple choice questions with four choices with one correct answer each. If we just randomly guess on each of the 5 questions, what is the probability that you get at least 1 question correct
The Correct Answer and Explanation is:
To find the probability that you get at least one question correct when randomly guessing on each of the 5 multiple choice questions, let’s break it down:
Step 1: Understand the setup
- Each question has 4 answer choices, with one correct answer.
- The probability of guessing correctly on one question is:
[
P(\text{correct}) = \frac{1}{4}
] - The probability of guessing incorrectly on one question is:
[
P(\text{incorrect}) = 1 – \frac{1}{4} = \frac{3}{4}
] - There are 5 questions, so you are making 5 independent guesses.
Step 2: Calculate the probability of getting no questions correct
To find the probability of getting at least one question correct, it’s easier to first calculate the probability of the complementary event—getting no questions correct.
The probability of getting one question incorrect is ( \frac{3}{4} ). For all 5 questions, since they are independent, the probability of getting all 5 questions incorrect is:
[
P(\text{all incorrect}) = \left( \frac{3}{4} \right)^5
]
Now, calculate this value:
[
\left( \frac{3}{4} \right)^5 = \frac{243}{1024} \approx 0.2373
]
So, the probability of getting all 5 questions incorrect is about 0.2373.
Step 3: Calculate the probability of getting at least one question correct
The probability of getting at least one question correct is the complement of the probability of getting all questions incorrect:
[
P(\text{at least 1 correct}) = 1 – P(\text{all incorrect}) = 1 – \frac{243}{1024} = \frac{781}{1024}
]
This simplifies to approximately:
[
P(\text{at least 1 correct}) \approx 0.7627
]
Final Answer
The probability of getting at least one question correct when guessing randomly on all 5 questions is approximately 0.7627 or 76.27%.
Explanation
In probability, the complement rule is often used when calculating the probability of “at least one” event happening. In this case, instead of calculating the probability of getting one, two, or more questions correct, we first calculate the easier-to-compute probability of getting zero questions correct and subtract that from 1. This approach simplifies the problem and leads us to the correct answer.