a. To find the probability of a student randomly guessing and getting exactly six questions correct, we need to determine the probability of getting one correct answer and multiply it by the number of ways the student can choose six questions out of the total ten.
The probability of getting one correct answer is 1/4, since there are four answer choices for each question. Therefore, the probability of getting one question correct is 1/4.
Now, we need to calculate the number of ways the student can choose six questions out of the total ten. This can be done using the combination formula, also known as "n choose k", which is given by:
nCk = n! / (k!(n-k)!)
In this case, we have n = 10 (total number of questions) and k = 6 (number of questions to be chosen).
Therefore, the number of ways to choose six questions out of ten is calculated as:
10C6 = 10! / (6!(10-6)!) = 10! / (6!4!) = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1) = 210.
Finally, we can find the probability of getting exactly six questions correct by multiplying the probability of each correct answer (1/4) by the number of ways to choose six questions out of ten (210):
Probability of getting exactly six questions correct = (1/4) * 210 = 210/4 = 52.5%.
b. No, getting exactly 10 questions correct is not the same probability as getting exactly zero correct.
To understand why, let's consider the probability of getting exactly zero questions correct. In order for this to happen, the student would need to guess the wrong answer for each of the ten questions.
Since there are four answer choices for each question, the probability of guessing the wrong answer for a single question is 3/4 (since there are three wrong choices out of four options).
Therefore, the probability of getting exactly zero questions correct would be: (3/4)^10 ≈ 0.056.
In contrast, getting exactly ten questions correct would mean guessing all the correct answers, which would have a probability of (1/4)^10 ≈ 0.0000009537.
As we can see, the probabilities are not the same, indicating that the chances of getting all questions correct or all questions wrong are very low compared to other possible outcomes.
c. To calculate the probability of getting at least six questions correct, assuming the student randomly guesses, we need to consider all possible outcomes where the number of correct questions is six or more.
Here are the steps to calculate this probability:
1. Calculate the probability of getting exactly six questions correct, following the steps described in part (a).
2. Repeat the process to calculate the probability of getting exactly seven, eight, nine, and ten questions correct.
3. Add up the probabilities obtained in step 2 to get the probability of getting at least six questions correct. This can be represented using the notion of "or" as follows:
Probability of getting at least six questions correct = Probability of getting exactly six questions correct
+ Probability of getting exactly seven questions correct
+ Probability of getting exactly eight questions correct
+ Probability of getting exactly nine questions correct
+ Probability of getting exactly ten questions correct.
Note that you do not need to actually calculate these probabilities unless specifically asked to. The steps provided above outline how you would approach finding the probability of getting at least six questions correct if the student randomly guesses.