student is taking up midterm exam for 3 minutes left to answer 10 remaining question of which there are 4 choices of answers for each question with only one choice correct. what is the probability of getting more than 5 of his answer is correct
To calculate the probability of getting more than 5 answers correct, we need to know the total number of ways to answer the remaining 10 questions and the number of ways to answer more than 5 correctly.
First, let's calculate the total number of ways to answer the remaining 10 questions. Each question has 4 choices, so there are 4 possibilities for the first question, 4 possibilities for the second question, and so on. Since all questions are independent and have the same number of choices, we can use the multiplication rule. Therefore, the total number of ways to answer the 10 questions is 4^10.
To calculate the number of ways to answer more than 5 questions correctly, we need to consider different scenarios:
- Choosing exactly 6 correct answers: There are 10 ways to choose which 6 questions the student gets correct (using binomial coefficient formula, C(10, 6)). For each of those 6 questions, there is 1 way to choose the correct answer and 3 ways to choose the wrong answer for the remaining 4 questions. So, the total number of ways to get exactly 6 correct answers is C(10, 6) * 1^6 * 3^4.
- Choosing exactly 7 correct answers: Similar to the previous case, there are C(10, 7) ways to choose which 7 questions the student gets correct. For each of those 7 questions, there is 1 way to choose the correct answer and 3 ways to choose the wrong answer for the remaining 3 questions. So, the total number of ways to get exactly 7 correct answers is C(10, 7) * 1^7 * 3^3.
- Choosing exactly 8, 9, or 10 correct answers: The calculation follows the same pattern as above, with C(10, 8), C(10, 9), and C(10, 10) respectively for the number of ways to choose the correctly answered questions. For each of these cases, the correct answers are chosen in 1 way, and the wrong answers for the remaining questions are chosen in 3 ways.
After calculating the total number of ways to answer more than 5 questions correctly for each case, we sum them up and divide it by the total number of ways to answer the 10 questions (4^10). So, the probability of getting more than 5 answers correct is:
( Total number of ways to answer more than 5 correctly ) / ( Total number of ways to answer the 10 questions )
P(more than 5 answers correct) = (C(10, 6) * 1^6 * 3^4 + C(10, 7) * 1^7 * 3^3 + C(10, 8) * 1^8 * 3^2 + C(10, 9) * 1^9 * 3^1 + C(10, 10) * 1^10 * 3^0) / (4^10)