A multiple choice test has 10 questions. Each question has four answer choices.
a. What is the probability a student randomly guesses the answers and gets exactly six questions correct?
b. Is getting exactly 10 questions correct the same probability as getting exactly zero correct? Explain.
c. Describe the steps needed to calculate the probability of getting at least six questions correct if the student randomly guesses. You do not need to calculate this probability
this is a binary probability ... correct (c) or wrong (w)
four possible answers ... ten questions
... p(c) = 1/4 = .25 ... p(w) = 3/4 = .75
a. (c + w)^10 = c^10 + 10 c^9 w + 45 c^8 w^2 + 120 c^7 w^3 + 210 c^6 w^4
... the 5th term is the probability of 6 correct
... p(6c) = 210 * .25^6 * .75^4 ≈ .016
b. no ... wrong answers are 3 times as likely as correct ones
... p(10c) = .25^10 ≈ .00000095
... p(10w) = .75^10 ≈ .056
c.. see a.
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.
a. To calculate the probability of a student randomly guessing and getting exactly six questions correct, we need to use the binomial probability formula.
The binomial probability formula is:
P(x) = (nCx) * p^x * q^(n-x)
Where:
- P(x) is the probability of getting x successes,
- n is the total number of trials (10 questions in this case),
- x is the number of successful outcomes (6 questions correct in this case),
- p is the probability of success on a single trial (1/4, assuming equal chance for each answer choice),
- q is the probability of failure on a single trial (3/4, as there are three incorrect answer choices).
Plugging in the values, we get:
P(6) = (10C6) * (1/4)^6 * (3/4)^(10-6)
To calculate this, we need to evaluate the combinations (C) value, which can be calculated using the formula:
nCr = n! / (r!(n-r)!), where n! means n factorial.
Once the combinations value is calculated, we can substitute it back into the binomial probability formula to find the answer.
b. The probability of getting exactly 10 questions correct is not the same as getting exactly zero correct. Getting exactly zero correct means that the student answered all questions incorrectly, while getting exactly ten correct means that the student answered all questions correctly. These two scenarios represent opposite outcomes, so their probabilities will be different.
c. To calculate the probability of getting at least six questions correct if the student randomly guesses, we need to consider all possible outcomes that provide at least six correct answers.
The steps to calculate this probability are as follows:
1. Calculate the probability of getting exactly six correct answers using the binomial probability formula (as explained in part a).
2. Calculate the probability of getting exactly seven correct answers using the same formula.
3. Repeat step 2 for eight correct answers, nine correct answers, and ten correct answers.
4. Add up the probabilities from steps 1-4 to get the probability of getting at least six questions correct.
Note that as the number of correct answers increases, the probabilities will become smaller.