A test is composed of six multiple choice questions where each question has 4 choices. If the answer choices for each question are equally likely, find the probability of answering more than 4 questions correctly
Use the binomial distribution, with parameters 6 and 1/4. Add up the probabilities for x=5 and x=6.
The probability of answering 5/6 = (1/4)^5 * 3/4
The probability of answering all 6 = (1/4)^6
Either-or probabilities are found by adding the separate probabilities.
To find the probability of answering more than 4 questions correctly, we need to know the total number of possible outcomes and the number of favorable outcomes.
The total number of possible outcomes is determined by the number of choices for each question. Since each question has 4 choices, there are 4 × 4 × 4 × 4 × 4 × 4 = 4^6 = 4096 possible outcomes for the test.
Now, let's find the number of favorable outcomes. To answer more than 4 questions correctly, we need to answer 5 or 6 questions correctly.
For 5 correct answers:
There are 6 questions, and we need to choose 5 of them to answer correctly. The number of ways to choose 5 questions out of 6 is given by the combination formula: C(6, 5) = 6!/5!(6-5)! = 6/1 = 6.
For each of these 6 combinations, there is only 1 correct way to answer the 5 chosen questions correctly.
For 6 correct answers:
There is only 1 way to answer all 6 questions correctly.
Therefore, the number of favorable outcomes is 6 + 1 = 7.
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
P(more than 4 questions correct) = Number of favorable outcomes / Total number of possible outcomes = 7/4096 ≈ 0.00171 (rounded to 5 decimal places)
So, the probability of answering more than 4 questions correctly is approximately 0.00171.