A student takes true or false test consisting of 10 questions by just guessing at each answer. Find the probability that the student's score will be 80% or more?
find the prob that he/she will get 8 out of 10, 9 out of 10, or 10 out of 10 correct
I will do the 8 out of 10, you do the others
prob(exactly 8 out of 10)
= C(10,8) (1/2)^8 (1/2)^2
= 45 (1/2)^10
= 45/1024
= appr .0439
Find the other two, and add them up
Well, let's see. Since there are only two choices for each question, the student has a 50% chance of guessing the correct answer on each question.
Now, the probability of getting 80% or more correct means getting 8 or more questions right.
The probability of getting exactly 8 questions right can be calculated using the binomial probability formula. It is (10 choose 8) * (0.5)^8 * (0.5)^2.
Similarly, we can calculate the probabilities of getting exactly 9 or exactly 10 questions right using the same formula.
Finally, we add up all these probabilities to get the total probability of getting 80% or more correct.
Or you could just give the student a high five for being lucky and suggest they try their hand at the lottery!
To find the probability of a student's score being 80% or more on a true or false test of 10 questions, we need to consider the different possibilities.
The student can score 8, 9, or 10 questions correctly to achieve an 80% or higher score.
1. Scoring 8 questions correctly:
The probability of guessing a question correctly is 1/2 (50% chance).
The probability of guessing a question incorrectly is also 1/2 (50% chance).
To score 8 questions correctly, the student needs to guess correctly on 8 questions and incorrectly on 2 questions.
Therefore, the probability of scoring 8 questions correctly is: (1/2)^8 * (1/2)^2 = (1/256) * (1/4) = 1/1024.
2. Scoring 9 questions correctly:
The probability of scoring 9 questions correctly is similar to scoring 8 questions correctly, but with one additional correct guess.
The student needs to guess correctly on 9 questions and incorrectly on 1 question.
Therefore, the probability of scoring 9 questions correctly is: (1/2)^9 * (1/2)^1 = (1/512) * (1/2) = 1/1024.
3. Scoring all 10 questions correctly:
The probability of scoring all 10 questions correctly is the same as guessing correctly on all 10 questions.
Therefore, the probability of scoring all 10 questions correctly is: (1/2)^10 = 1/1024.
Now, we can sum up the probabilities of the different possible scores (8, 9, or 10 questions correct):
1/1024 + 1/1024 + 1/1024 = 3/1024.
Therefore, the probability of the student's score being 80% or more is 3/1024.
To find the probability that the student's score will be 80% or more, we need to determine the number of questions the student needs to answer correctly in order to achieve this score.
Since the test consists of 10 true or false questions, the most number of questions the student can answer correctly by guessing is 10.
To calculate the probability, we can consider each possible score (i.e., the number of questions answered correctly) and sum their probabilities.
Let's go step by step:
Step 1: Determine the number of questions the student needs to answer correctly to achieve at least an 80% score.
Since 80% of 10 questions is 8 questions, the student needs to answer 8 or more questions correctly.
Step 2: Calculate the probability of answering exactly 8, 9, or 10 questions correctly.
Since each question has two possible answers (true or false), and the student is guessing, the probability of guessing the correct answer for each question is 1/2 or 0.5.
The probability of answering exactly k questions correctly out of 10 can be calculated using the binomial probability formula:
P(k) = C(n, k) * p^k * q^(n-k)
Where:
P(k) is the probability of answering exactly k questions correctly,
C(n, k) is the binomial coefficient (n choose k),
p is the probability of guessing the correct answer (0.5),
q is the probability of guessing the incorrect answer (0.5),
n is the total number of questions (10), and
k is the number of questions answered correctly.
Step 3: Sum the probabilities of answering at least 8, 9, or 10 questions correctly.
P(8) + P(9) + P(10)
Now, let's calculate the probabilities:
P(8) = C(10, 8) * 0.5^8 * 0.5^2 = 45 * 0.00390625 ≈ 0.17578125
P(9) = C(10, 9) * 0.5^9 * 0.5^1 = 10 * 0.001953125 ≈ 0.01953125
P(10) = C(10, 10) * 0.5^10 * 0.5^0 = 1 * 0.0009765625 ≈ 0.0009765625
Step 4: Sum the probabilities:
P(score ≥ 80%) = P(8) + P(9) + P(10) ≈ 0.17578125 + 0.01953125 + 0.0009765625 ≈ 0.1962890625
Therefore, the probability that the student's score will be 80% or more is approximately 0.196 or 19.6%.