A true/ false test is given. If a person guesses the answers, the probability that any particular question is correctly answered is 0.5. If the test contains 14 questions and 7 correct answers is a pass, what is the probability of passing the test by simply guessing?
1/2
To determine the probability of passing the test by simply guessing, we need to calculate the probability of getting 7 or more questions correct out of 14.
First, we need to identify the probability of getting exactly 7 questions correct. Since each question has two possible answers (true or false), the probability of guessing any particular question correctly is 0.5. So, the probability of guessing 7 questions correctly can be calculated using the binomial probability formula:
P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
P(X=k) is the probability of getting k successes,
C(n, k) is the number of combinations of n items taken k at a time,
p is the probability of success on a single trial,
(1-p) is the probability of failure on a single trial,
n is the number of trials (14 questions in this case), and
k is the number of successes (7 questions answered correctly).
Plugging in the values, we get:
P(X=7) = C(14, 7) * 0.5^7 * (1-0.5)^(14-7)
Calculating the values, we have:
P(X=7) = 3003 * 0.5^7 * 0.5^7 = 3003 * 0.5^14
Next, we need to calculate the probabilities for getting 8, 9, 10, 11, 12, 13, and 14 questions correctly.
P(X=8) = C(14, 8) * 0.5^8 * 0.5^6
P(X=9) = C(14, 9) * 0.5^9 * 0.5^5
P(X=10) = C(14, 10) * 0.5^10 * 0.5^4
P(X=11) = C(14, 11) * 0.5^11 * 0.5^3
P(X=12) = C(14, 12) * 0.5^12 * 0.5^2
P(X=13) = C(14, 13) * 0.5^13 * 0.5^1
P(X=14) = C(14, 14) * 0.5^14 * 0.5^0
Now, we can calculate the probability of passing by summing up these individual probabilities:
P(passing) = P(X=7) + P(X=8) + P(X=9) + P(X=10) + P(X=11) + P(X=12) + P(X=13) + P(X=14)
By adding up these probabilities, we can determine the final probability of passing the test by simply guessing.
To solve this problem, we can use the concept of binomial probability.
In this case, each question has two possible outcomes: a correct answer or an incorrect answer. The probability of guessing the correct answer for any particular question is 0.5.
Now, let's calculate the probability of passing the test by simply guessing.
We know that to pass the test, at least 7 out of 14 questions must be answered correctly.
The probability of passing the test can be calculated by summing the individual probabilities of getting 7, 8, 9, ..., 14 questions correct.
Using the binomial probability formula, the probability of getting exactly k questions correct out of n questions is:
P(k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
- (n C k) represents the number of ways to choose k items from a set of n items (also known as combinations), calculated as n! / (k! * (n - k)!)
- p is the probability of success (getting a question correct)
- n is the total number of questions
We can calculate the probability of passing the test as follows:
P(pass) = P(7) + P(8) + P(9) + P(10) + P(11) + P(12) + P(13) + P(14)
P(pass) = [(14 C 7) * 0.5^7 * (1 - 0.5)^(14 - 7)] + [(14 C 8) * 0.5^8 * (1 - 0.5)^(14 - 8)] + [(14 C 9) * 0.5^9 * (1 - 0.5)^(14 - 9)] + [(14 C 10) * 0.5^10 * (1 - 0.5)^(14 - 10)] + [(14 C 11) * 0.5^11 * (1 - 0.5)^(14 - 11)] + [(14 C 12) * 0.5^12 * (1 - 0.5)^(14 - 12)] + [(14 C 13) * 0.5^13 * (1 - 0.5)^(14 - 13)] + [(14 C 14) * 0.5^14 * (1 - 0.5)^(14 - 14)]
Calculating these values will give us the probability of passing the test by simply guessing.