The following is a table of probabilties calculated from a survey of BHCC students with the question asked "How many classes are you taking this semester?"
x: # of classes 1 2 3 4 5
P(x) 0.23 0.27 0.13 0.33 0.04
Using the table, find the following probabilities for a student selected at random:
a.) What is the probability that a student is taking 2 or more classes?
b.) What is the probability that a student is taking at least 3 classes?
c.) What is the probability that a student is taking more than 3 classes?
d.) What is the probability that a student is taking less than 2 classes?
e.) What is the probability that a student is taking no more than 2 clases?
f.) What is the average (mean) amount of classes a student takes at BHCC?
g.) What is the standard deviation for the amount of classes a student takes at BHCC?
a.) The probability that a student is taking 2 or more classes is equal to the sum of the probabilities for taking 2, 3, 4, and 5 classes. So, P(2 or more classes) = P(2) + P(3) + P(4) + P(5) = 0.27 + 0.13 + 0.33 + 0.04 = 0.77.
b.) The probability that a student is taking at least 3 classes is equal to the sum of the probabilities for taking 3, 4, and 5 classes. So, P(at least 3 classes) = P(3) + P(4) + P(5) = 0.13 + 0.33 + 0.04 = 0.50.
c.) The probability that a student is taking more than 3 classes is equal to the sum of the probabilities for taking 4 and 5 classes. So, P(more than 3 classes) = P(4) + P(5) = 0.33 + 0.04 = 0.37.
d.) The probability that a student is taking less than 2 classes is equal to the sum of the probabilities for taking 1 class. So, P(less than 2 classes) = P(1) = 0.23.
e.) The probability that a student is taking no more than 2 classes is equal to the sum of the probabilities for taking 1 and 2 classes. So, P(no more than 2 classes) = P(1) + P(2) = 0.23 + 0.27 = 0.50.
f.) To find the average (mean) amount of classes a student takes at BHCC, we need to multiply each number of classes by its respective probability and sum them up. So, (1 * 0.23) + (2 * 0.27) + (3 * 0.13) + (4 * 0.33) + (5 * 0.04) = 1.23. Thus, the average amount of classes a student takes at BHCC is 1.23.
g.) To find the standard deviation for the amount of classes a student takes at BHCC, we'll need to calculate the variance first. The variance can be found using the formula: Var(X) = Σ [ (x - μ)^2 * P(x) ], where μ is the mean.
So, the variance for the amount of classes can be calculated as: ( (1 - 1.23)^2 * 0.23 ) + ( (2 - 1.23)^2 * 0.27 ) + ( (3 - 1.23)^2 * 0.13 ) + ( (4 - 1.23)^2 * 0.33 ) + ( (5 - 1.23)^2 * 0.04 ) = 1.0661.
The standard deviation is the square root of the variance. So, the standard deviation for the amount of classes a student takes at BHCC is approximately √1.0661 ≈ 1.03.
a.) To find the probability that a student is taking 2 or more classes, add the probabilities for taking 2, 3, 4, and 5 classes:
P(2 or more classes) = P(2) + P(3) + P(4) + P(5)
= 0.27 + 0.13 + 0.33 + 0.04
= 0.77
Therefore, the probability that a student is taking 2 or more classes is 0.77.
b.) To find the probability that a student is taking at least 3 classes, add the probabilities for taking 3, 4, and 5 classes:
P(at least 3 classes) = P(3) + P(4) + P(5)
= 0.13 + 0.33 + 0.04
= 0.5
Therefore, the probability that a student is taking at least 3 classes is 0.5.
c.) To find the probability that a student is taking more than 3 classes, add the probabilities for taking 4 and 5 classes:
P(more than 3 classes) = P(4) + P(5)
= 0.33 + 0.04
= 0.37
Therefore, the probability that a student is taking more than 3 classes is 0.37.
d.) To find the probability that a student is taking less than 2 classes, add the probabilities for taking 1 class:
P(less than 2 classes) = P(1)
= 0.23
Therefore, the probability that a student is taking less than 2 classes is 0.23.
e.) To find the probability that a student is taking no more than 2 classes, add the probabilities for taking 1 and 2 classes:
P(no more than 2 classes) = P(1) + P(2)
= 0.23 + 0.27
= 0.5
Therefore, the probability that a student is taking no more than 2 classes is 0.5.
f.) To calculate the average (mean) amount of classes a student takes at BHCC, multiply each number of classes by its corresponding probability and sum the results:
Mean = (1 * 0.23) + (2 * 0.27) + (3 * 0.13) + (4 * 0.33) + (5 * 0.04)
= 0.23 + 0.54 + 0.39 + 1.32 + 0.2
= 2.68
Therefore, the average (mean) amount of classes a student takes at BHCC is 2.68.
g.) To calculate the standard deviation for the amount of classes a student takes at BHCC, you can use the formula:
Standard Deviation = sqrt(((1-mean)^2 * P(1)) + ((2-mean)^2 * P(2)) + ((3-mean)^2 * P(3)) + ((4-mean)^2 * P(4)) + ((5-mean)^2 * P(5)))
Substituting the values from the table:
Standard Deviation = sqrt(((1-2.68)^2 * 0.23) + ((2-2.68)^2 * 0.27) + ((3-2.68)^2 * 0.13) + ((4-2.68)^2 * 0.33) + ((5-2.68)^2 * 0.04))
You can calculate this to get the standard deviation for the amount of classes a student takes at BHCC.
To find the probabilities for the given questions, we will use the provided table.
a) To find the probability that a student is taking 2 or more classes, we sum up the probabilities for taking 2, 3, 4, and 5 classes:
P(2 or more) = P(2) + P(3) + P(4) + P(5)
= 0.27 + 0.13 + 0.33 + 0.04
= 0.77
Therefore, the probability that a student is taking 2 or more classes is 0.77.
b) To find the probability that a student is taking at least 3 classes, we sum up the probabilities for taking 3, 4, and 5 classes:
P(at least 3) = P(3) + P(4) + P(5)
= 0.13 + 0.33 + 0.04
= 0.50
Therefore, the probability that a student is taking at least 3 classes is 0.50.
c) To find the probability that a student is taking more than 3 classes, we sum up the probabilities for taking 4 and 5 classes:
P(more than 3) = P(4) + P(5)
= 0.33 + 0.04
= 0.37
Therefore, the probability that a student is taking more than 3 classes is 0.37.
d) To find the probability that a student is taking less than 2 classes, we sum up the probabilities for taking 1 class:
P(less than 2) = P(1)
= 0.23
Therefore, the probability that a student is taking less than 2 classes is 0.23.
e) To find the probability that a student is taking no more than 2 classes, we sum up the probabilities for taking 1 and 2 classes:
P(no more than 2) = P(1) + P(2)
= 0.23 + 0.27
= 0.50
Therefore, the probability that a student is taking no more than 2 classes is 0.50.
f) To find the average (mean) amount of classes a student takes at BHCC, we multiply each value of x (number of classes) by its corresponding probability and sum up the results:
Mean = (1 * P(1)) + (2 * P(2)) + (3 * P(3)) + (4 * P(4)) + (5 * P(5))
= (1 * 0.23) + (2 * 0.27) + (3 * 0.13) + (4 * 0.33) + (5 * 0.04)
= 0.23 + 0.54 + 0.39 + 1.32 + 0.20
= 2.68
Therefore, the average (mean) amount of classes a student takes at BHCC is 2.68.
g) To find the standard deviation for the amount of classes a student takes at BHCC, we first need to calculate the variance. The variance is the average of the squared differences between each value of x and the mean.
Variance = [(1 - Mean)^2 * P(1)] + [(2 - Mean)^2 * P(2)] + [(3 - Mean)^2 * P(3)] + [(4 - Mean)^2 * P(4)] + [(5 - Mean)^2 * P(5)]
= [(1 - 2.68)^2 * 0.23] + [(2 - 2.68)^2 * 0.27] + [(3 - 2.68)^2 * 0.13] + [(4 - 2.68)^2 * 0.33] + [(5 - 2.68)^2 * 0.04]
= [(-1.68)^2 * 0.23] + [(-0.68)^2 * 0.27] + [(0.32)^2 * 0.13] + [(1.32)^2 * 0.33] + [(2.32)^2 * 0.04]
= [2.8224 * 0.23] + [0.4624 * 0.27] + [0.1024 * 0.13] + [1.7424 * 0.33] + [5.3824 * 0.04]
= 0.6488 + 0.1248 + 0.0133 + 0.5748 + 0.2153
= 1.5770
Now, we can calculate the standard deviation by taking the square root of the variance:
Standard Deviation = sqrt(Variance)
= sqrt(1.5770)
≈ 1.2567
Therefore, the standard deviation for the amount of classes a student takes at BHCC is approximately 1.2567.