Professor Sanchez has been teaching Principles of Economics for over 25 years. He uses the following scale for grading.
Grade Numerical Score Probability
A 4 0.120
B 3 0.210
C 2 0.400
D 1 0.155
F 0 0.115
a. Convert the above probability distribution to a cumulative probability distribution. (Round your answers to 3 decimal places.)
b. What is the probability of earning at least a B in Professor Sanchez’s course? (Round your answer to 3 decimal places.)
c. What is the probability of passing Professor Sanchez’s course? (Round your answer to 3 decimal places.)
a. To convert the probability distribution to a cumulative probability distribution, we need to add up the probabilities for each grade up to that point.
Grade Numerical Score Probability Cumulative Probability
A 4 0.120 0.120
B 3 0.210 0.120 + 0.210 = 0.330
C 2 0.400 0.330 + 0.400 = 0.730
D 1 0.155 0.730 + 0.155 = 0.885
F 0 0.115 0.885 + 0.115 = 1.000
b. The probability of earning at least a B is the cumulative probability for a B or higher. From the table in part a, the cumulative probability for a B is 0.330. Therefore, the probability of earning at least a B is 0.330.
c. The probability of passing the course is the cumulative probability for a C or higher. From the table in part a, the cumulative probability for a C is 0.730. Therefore, the probability of passing the course is 0.730.
a. To convert the probability distribution to a cumulative probability distribution, we need to find the cumulative probabilities for each grade. The cumulative probability is the sum of the probabilities of all the previous grades.
To calculate the cumulative probabilities, we can start with the first grade and add the probability of that grade to the cumulative probability.
Here is the calculation for each grade:
Grade Numerical Score Probability Cumulative Probability
A 4 0.120 0.120
B 3 0.210 0.120 + 0.210 = 0.330
C 2 0.400 0.330 + 0.400 = 0.730
D 1 0.155 0.730 + 0.155 = 0.885
F 0 0.115 0.885 + 0.115 = 1.000
b. To find the probability of earning at least a B, we need to sum the probabilities of getting a B, A, or any higher grade.
The cumulative probability for a B or higher is the cumulative probability for A (0.120) plus the cumulative probability for B (0.330).
Probability of earning at least a B = 0.120 + 0.330 = 0.450
So the probability of earning at least a B in Professor Sanchez’s course is 0.450.
c. To find the probability of passing Professor Sanchez's course, we need to sum the probabilities of getting a C, B, or A.
The cumulative probability for passing is the cumulative probability for C (0.730) plus the cumulative probability for B (0.330) plus the cumulative probability for A (0.120).
Probability of passing = 0.730 + 0.330 + 0.120 = 1.180
However, since the cumulative probability cannot exceed 1, we round this down to 1.000.
So the probability of passing Professor Sanchez’s course is 1.000.
a. To convert the probability distribution to a cumulative probability distribution, we need to calculate the cumulative probabilities for each grade.
To do this, we can start by listing the grades in order from highest to lowest numerical score and calculate the cumulative probabilities at each step.
Grade | Numerical Score | Probability | Cumulative Probability
A | 4 | 0.120 |
B | 3 | 0.210 |
C | 2 | 0.400 |
D | 1 | 0.155 |
F | 0 | 0.115 |
Now, let's calculate the cumulative probabilities.
For the first grade, A, its cumulative probability is simply the probability of A, which is 0.120.
For the second grade, B, its cumulative probability is the sum of the probability of B and the cumulative probability of A, which is 0.210 + 0.120 = 0.330.
For the third grade, C, its cumulative probability is the sum of the probability of C and the cumulative probability of B, which is 0.400 + 0.330 = 0.730.
For the fourth grade, D, its cumulative probability is the sum of the probability of D and the cumulative probability of C, which is 0.155 + 0.730 = 0.885.
For the last grade, F, its cumulative probability is the sum of the probability of F and the cumulative probability of D, which is 0.115 + 0.885 = 1.000.
So, the cumulative probability distribution is:
Grade | Numerical Score | Probability | Cumulative Probability
A | 4 | 0.120 | 0.120
B | 3 | 0.210 | 0.330
C | 2 | 0.400 | 0.730
D | 1 | 0.155 | 0.885
F | 0 | 0.115 | 1.000
b. To calculate the probability of earning at least a B in Professor Sanchez’s course, we need to find the cumulative probability of B or higher.
Looking at the cumulative probability distribution, we can see that the cumulative probability for B is 0.330.
Therefore, the probability of earning at least a B is 0.330.
c. To calculate the probability of passing Professor Sanchez’s course, we need to find the cumulative probability of C or higher.
Looking at the cumulative probability distribution, we can see that the cumulative probability for C is 0.730.
Therefore, the probability of passing Professor Sanchez’s course is 0.730.