A student randomly guessed the answers to a five question true/false test (there is a 50% chance of guessing correctly on each question). Find.
1. the probability that the student answers more than 1 question correctly
2. the expected number of correct answers
3.the standard deviation
PLEASE HELP!!!
This is a binomial distribution. A tree diagram or 2^n will give you the sample space of 32. Then:
1. P(X>1) = 1 - P(X=0) - P(X=1)
= 1 - 1/32 - 5/32
= ?
2. Mean = np
= (5)(0.5)
= ?
3. SD = sqrt(np(1-p)
= ?
1. To find the probability that the student answers more than 1 question correctly, we can use the concept of binomial probability.
The probability of answering a single question correctly through guessing is 1/2 or 0.5 since there are only two possible outcomes: true or false.
To find the probability of answering exactly k questions correctly out of the five, we use the binomial probability formula:
P(X = k) = (nCk) * p^k * q^(n-k)
Where:
- P(X = k) is the probability of getting exactly k successes (correct answers)
- n is the total number of trials (number of questions)
- k is the number of successes (correct answers)
- p is the probability of success (probability of guessing correctly)
- q is the probability of failure (probability of guessing incorrectly)
In this case, n = 5 (five questions) and p = 0.5 (probability of guessing correctly)
Now, let's calculate the probability of the student answering more than 1 question correctly.
P(X > 1) = P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
P(X = 2) = (5C2) * (0.5)^2 * (0.5)^(5-2)
P(X = 3) = (5C3) * (0.5)^3 * (0.5)^(5-3)
P(X = 4) = (5C4) * (0.5)^4 * (0.5)^(5-4)
P(X = 5) = (5C5) * (0.5)^5 * (0.5)^0
To calculate the binomial coefficients (nCk), you can use the formula:
nCk = n! / (k! * (n-k)!)
Now plugging in the values and calculating:
P(X = 2) = (5C2) * (0.5)^2 * (0.5)^3 = 10 * 0.25 * 0.125 = 0.3125
P(X = 3) = (5C3) * (0.5)^3 * (0.5)^2 = 10 * 0.125 * 0.25 = 0.3125
P(X = 4) = (5C4) * (0.5)^4 * (0.5)^1 = 5 * 0.0625 * 0.5 = 0.15625
P(X = 5) = (5C5) * (0.5)^5 * (0.5)^0 = 1 * 0.03125 * 1 = 0.03125
Now, summing up the probabilities:
P(X > 1) = 0.3125 + 0.3125 + 0.15625 + 0.03125 = 0.8125
So, the probability that the student answers more than 1 question correctly is 0.8125 or 81.25%.
2. To find the expected number of correct answers, we can multiply the probability of getting each possible number of correct answers by that number, and then sum them up.
Expected number of correct answers = (0 * P(X = 0)) + (1 * P(X = 1)) + (2 * P(X = 2)) + (3 * P(X = 3)) + (4 * P(X = 4)) + (5 * P(X = 5))
Using the previously calculated probabilities, we get:
Expected number of correct answers = (0 * 0.03125) + (1 * 0.15625) + (2 * 0.3125) + (3 * 0.3125) + (4 * 0.15625) + (5 * 0.03125)
= 0.15625 + 0.625 + 0.9375 + 0.625 + 0.625 + 0.15625
= 3.125
Therefore, the expected number of correct answers is 3.125.
3. To calculate the standard deviation, we need to use the formula:
Standard deviation = sqrt(n * p * q)
Where:
- n is the total number of trials (number of questions)
- p is the probability of success (probability of guessing correctly)
- q is the probability of failure (probability of guessing incorrectly)
In this case, n = 5 (five questions) and p = 0.5 (probability of guessing correctly)
Standard deviation = sqrt(5 * 0.5 * 0.5)
= sqrt(1.25)
≈ 1.118
Therefore, the standard deviation is approximately 1.118.