Giving a test to a group of students, the grades and gender are summarized below. Round your answers to 4 decimal places. Grades and Gender A В Total Male 14 4 12 30 Female 7 9. 3 19 Total 21 13 15 49 If one student is chosen at random, a. Find the probability that the student got a C: 0.3061 b. Find the probability that the student was female AND got a "C": 0.0612 c. Find the probability that the student was male OR got a "C": d. If one student is chosen at random, find the probability that the student was female GIVEN they got a 'C':
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
![### Probabilities of Student Grades by Gender
#### Data Summary
A test was administered to a group of students, and the results are categorized by grades (A, B, C) and gender (Male, Female). The following table summarizes the data:
| Grades and Gender | A | B | C | Total |
|-------------------|----|----|----|-------|
| Male | 14 | 4 | 12 | 30 |
| Female | 7 | 9 | 3 | 19 |
| **Total** | 21 | 13 | 15 | 49 |
#### Probability Calculations
The task is to find the probabilities for various scenarios when a student is chosen at random. The answers must be rounded to four decimal places.
**a. Probability that a student got a C**
To find this probability, divide the total number of students who got a C by the total number of students:
\[
P(C) = \frac{15}{49} \approx 0.3061
\]
**b. Probability that the student was female AND got a C**
For this probability, consider the number of females who got a C divided by the total number of students:
\[
P(\text{Female and C}) = \frac{3}{49} \approx 0.0612
\]
**c. Probability that the student was male OR got a C**
This probability involves students who are either male, got a C, or both. Use the formula for the probability of A OR B:
\[
P(\text{Male or C}) = P(\text{Male}) + P(C) - P(\text{Male and C})
\]
Calculate each:
- \( P(\text{Male}) = \frac{30}{49} \)
- \( P(C) = \frac{15}{49} \)
- \( P(\text{Male and C}) = \frac{12}{49} \)
Plug these into the formula:
\[
P(\text{Male or C}) = \frac{30}{49} + \frac{15}{49} - \frac{12}{49} = \frac{33}{49} \approx 0.6735
\]
**d. Probability that the student was female GIVEN they got a C**
This is a conditional probability question. Calculate it by](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0017d92f-fd5a-4a34-8e16-36b3a20e3d6d%2F206d24fe-ed68-4797-8b80-882bfee42eff%2Fhqjyicd_processed.png&w=3840&q=75)
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