A student randomly guesses the answers to a four question true or false quiz. Find the probability that the student answers F on at least 2 of the questions
To find the probability that the student answers F on at least 2 of the questions, we can use a combination of counting favorable outcomes and total outcomes.
First, let's determine the total number of possible outcomes for each question. Since each question has two possible choices (true or false), there are 2 options for each question, resulting in a total of 2^4 = 16 possible outcomes.
Next, let's determine the number of favorable outcomes, i.e., the number of ways the student can answer F on at least 2 of the questions.
There are three scenarios to consider:
1. The student answers F on all four questions.
2. The student answers F on exactly three questions.
3. The student answers F on exactly two questions.
For each scenario, let's calculate the number of ways it can happen:
1. The student answers F on all four questions.
In this case, there is only one way the student can answer F on all four questions.
2. The student answers F on exactly three questions.
To calculate the number of ways the student can answer F on exactly three questions, we can choose 3 questions out of 4, and for each chosen question, the student answers F. This can be calculated using the combination formula: C(4, 3) = 4.
3. The student answers F on exactly two questions.
To calculate the number of ways the student can answer F on exactly two questions, we can choose 2 questions out of 4, and for each chosen question, the student answers F. This can be calculated using the combination formula: C(4, 2) = 6.
Now, let's add up the number of favorable outcomes:
1 (all four questions) + 4 (exactly three questions) + 6 (exactly two questions) = 11.
Therefore, the number of favorable outcomes is 11.
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
P(F on at least 2 questions) = number of favorable outcomes / total number of possible outcomes = 11 / 16 ≈ 0.6875.
So, the probability that the student answers F on at least 2 of the questions is approximately 0.6875, or about 68.75%.