A student takes a true-false test that has 8 questions and guesses randomly at each answer. Let X be the number of questions answered correctly. Find P(6 or more)
To find P(6 or more), we need to calculate the probability of getting 6, 7, or 8 questions correctly out of the 8 questions on the test.
To go step-by-step, let's calculate the probability of getting exactly 6 questions correct:
Step 1: Calculate the probability of getting a question correct: Since the student is guessing randomly, there are 2 possible outcomes for each question (true or false) and only 1 of those is correct. Therefore, the probability of getting a question correct is 1/2.
Step 2: Calculate the probability of getting 6 questions correct: Since there are 8 questions on the test, we need to choose 6 questions out of those 8 to answer correctly. The total number of ways to choose 6 questions out of 8 is given by the binomial coefficient (nCr) formula.
The probability of getting exactly k successes in n trials is given by the binomial probability formula:
P(X = k) = (nCr) * (p^k) * (1-p)^(n-k)
Where n is the total number of trials (8 questions), k is the number of successes (6 questions correct), nCr is the binomial coefficient, p is the probability of success (1/2), and (1-p) is the probability of failure (1 - 1/2 = 1/2).
Using the binomial probability formula, we can calculate the probability of getting exactly 6 questions correct:
P(X = 6) = (8C6) * (1/2)^6 * (1/2)^(8-6)
Step 3: Calculate the probabilities for getting 7 and 8 questions correct using the same approach as in step 2.
P(X = 7) = (8C7) * (1/2)^7 * (1/2)^(8-7)
P(X = 8) = (8C8) * (1/2)^8 * (1/2)^(8-8)
Step 4: Calculate the probability of getting 6 or more questions correct:
P(6 or more) = P(X = 6) + P(X = 7) + P(X = 8)
Now, let's calculate the values:
Step 1: The probability of getting a question correct is 1/2.
Step 2: Using the binomial coefficient formula, (8C6) = 8! / (6! * (8-6)!) = 28.
P(X = 6) = (8C6) * (1/2)^6 * (1/2)^(8-6) = 28 * (1/2)^6 * (1/2)^2 = 0.109375.
Step 3: (8C7) = 8! / (7! * (8-7)!) = 8.
P(X = 7) = (8C7) * (1/2)^7 * (1/2)^(8-7) = 8 * (1/2)^7 * (1/2)^1 = 0.03125.
(8C8) = 8! / (8! * (8-8)!) = 1.
P(X = 8) = (8C8) * (1/2)^8 * (1/2)^(8-8) = 1 * (1/2)^8 * (1/2)^0 = 0.00390625.
Step 4: P(6 or more) = P(X = 6) + P(X = 7) + P(X = 8) = 0.109375 + 0.03125 + 0.00390625 = 0.14453125.
Therefore, the probability of getting 6 or more questions correct is approximately 0.1445.
To find the probability of getting 6 or more questions correct, we need to calculate the individual probabilities of getting 6, 7, and 8 questions correct, and then sum them up.
Let's first calculate the probability of getting exactly 6 questions correct:
Since there are two options (true or false) for each question and the student guesses randomly, the probability of guessing the correct answer for any question is 1/2. Therefore, the probability of getting exactly 6 questions correct is (1/2)^6.
Next, let's calculate the probability of getting exactly 7 questions correct:
Similar to the previous calculation, the probability of guessing the correct answer for any question is 1/2. Therefore, the probability of getting exactly 7 questions correct is (1/2)^7.
Finally, we calculate the probability of getting exactly 8 questions correct:
The probability of getting the correct answer for any question is still 1/2. Therefore, the probability of getting exactly 8 questions correct is (1/2)^8.
To find P(6 or more), we need to sum up these probabilities:
P(6 or more) = P(6 correct) + P(7 correct) + P(8 correct)
= (1/2)^6 + (1/2)^7 + (1/2)^8
Now we can calculate P(6 or more) by plugging in the values:
P(6 or more) = (1/2)^6 + (1/2)^7 + (1/2)^8
= 1/64 + 1/128 + 1/256
= 1/64 + 1/64 + 1/64
= 3/64
Therefore, the probability of getting 6 or more questions correct is 3/64.
Find P(6 or more) implies
P(6) + P(7) + P(8)
= C(8,6)(1/2)^6 (1/2)^2 + C(8,7) (1/2)^8 + C(8,8) (1/2)^8
= (1/2)^8 (28+8+1) = 37/256