A true-false test consists of 15 items.
(a) If Chris does not study at all and guesses each and every item in the test, describe the probability model for the number of correct guesses.
(b) What is the probability that Chris gets 80% or more for the test?
Not sure how to do a). Not sure if i did b) correct.
b) he must get 12 right or 13 right or 14 right or 15 right
= C(15,12)(1/2)^12 (1/2)^2 + C(15,13)(1/2)^13(1/2) + C(15,14) (1/2)^14+C(15,15)(1/2)^15
= 0.0426%
(a) it is a normal distribution (bell curve)n with the most likely score being half right (or wrong)
you could plot the points using your binomial distribution from part (b)
(b) the exponents in an individual term should sum to 15
only your last term does that
otherwise, looks okay
To answer part (a) of the question, let's consider the probability model for the number of correct guesses when Chris does not study and guesses each and every item in the test.
In a true-false test, there are only two possible outcomes for each item - either Chris gets it correct or incorrect. Let's assume Chris guesses randomly, which means he has a 50% chance of getting each item correct and a 50% chance of getting it incorrect.
Now, let's define a random variable X to represent the number of correct guesses. X can take on values from 0 to 15, as there are 15 items in the test.
To find the probability model for X, we need to determine the probability of each possible outcome. Let's use the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- n is the number of trials (in this case, the number of items in the test, which is 15)
- k is the number of successes (number of correct guesses)
- p is the probability of success (probability of getting an item correct, which is 0.5)
Now, let's calculate the probabilities for each possible value of X:
P(X = 0) = C(15, 0) * (0.5^0) * (0.5^15) = 0.0000305
P(X = 1) = C(15, 1) * (0.5^1) * (0.5^14) = 0.000458
P(X = 2) = C(15, 2) * (0.5^2) * (0.5^13) = 0.003891
...
P(X = 15) = C(15, 15) * (0.5^15) * (0.5^0) = 0.0000305
So, the probability model for the number of correct guesses when Chris guesses randomly on a true-false test is:
X | P(X)
--------------
0 | 0.0000305
1 | 0.000458
2 | 0.003891
... | ...
15 | 0.0000305
Now, let's move on to part (b) of the question. You correctly calculated the probability of Chris getting 12, 13, 14, or 15 items right using the binomial probability formula. However, you made an error while calculating these probabilities.
To find the probability that Chris gets 80% or more on the test (which means getting 12, 13, 14, or 15 items correct), we need to sum up the probabilities of these events.
P(X >= 12) = P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)
Let's calculate these probabilities correctly:
P(X = 12) = C(15, 12) * (0.5^12) * (0.5^3) = 0.01709
P(X = 13) = C(15, 13) * (0.5^13) * (0.5^2) = 0.00354
P(X = 14) = C(15, 14) * (0.5^14) * (0.5^1) = 0.00027
P(X = 15) = C(15, 15) * (0.5^15) * (0.5^0) = 0.0000305
Now, let's add up these probabilities:
P(X >= 12) = 0.01709 + 0.00354 + 0.00027 + 0.0000305 = 0.0219
Therefore, the probability that Chris gets 80% or more (12 or more) items correct on the test is approximately 0.0219, which is about 2.19%.