You guess on all of the answers on a 10 question quiz. If there are 4 possible answers for each question, what is the probability that you get 3 questions right?
i've tried doing this problem multiple times and i'm just not understanding.. step by step would be very helpful.
prob of getting it right = 1/4
prob of getting it wrong = 3/4
prob of getting 3 of 10 right
= C(10,3) (1/4)^3 (3/4)^7
= 120(1/64)(2187/16384)
= appr .25
To find the probability of getting exactly 3 questions right, we need to calculate the following:
1. Determine the number of ways you can choose 3 out of the 10 questions to answer correctly.
2. For each of these choices, calculate the probability of getting that combination of correct answers.
Here's a step-by-step breakdown:
Step 1: Determine the number of ways to choose 3 questions out of 10
To calculate this, we use the combination formula. The number of ways to choose k items from a set of n items is given by:
nCr = n! / (r!(n-r)!)
In this case, n = 10 (total number of questions) and r = 3 (number of questions we want to answer correctly). Plugging these values in:
10C3 = 10! / (3!(10-3)!)= 10! / (3!7!)
Simplifying:
10C3 = (10 * 9 * 8) / (3 * 2 * 1) = 120
So, there are 120 different ways you can choose 3 questions to answer correctly.
Step 2: Calculate the probability for each combination
Since each question has 4 possible answers, the probability of guessing the correct answer for any one question is 1/4.
The probability of guessing incorrectly for any one question is 3/4.
To calculate the probability for each combination of 3 correctly guessed questions, we multiply the probabilities of guessing correctly 3 times and the probabilities of guessing incorrectly for the remaining 7 questions:
Probability of getting 3 questions right = (1/4)^3 * (3/4)^7
Calculating:
Probability of getting 3 questions right = 0.015625 * 0.1334838867 = 0.002088546
So, the probability of getting exactly 3 questions right when guessing on a 10-question quiz with 4 possible answers for each question is approximately 0.0021, or 0.21%.