To solve the problems, we first need to calculate the total number of students in each category and then use that information to find the probabilities.
Given Data:
- Freshman:
- Satisfied: 52
- Neutral: 26
- Not satisfied: 21
- Sophomore:
- Satisfied: 52
- Neutral: 20
- Not satisfied: 22
- Junior:
- Satisfied: 60
- Neutral: 20
- Not satisfied: 11
- Senior:
- Satisfied: 57
- Neutral: 10
- Not satisfied: 28
Total Students:
To find the total number of students surveyed, we sum all the students in each category:
[
\text{Total} = (52 + 26 + 21) + (52 + 20 + 22) + (60 + 20 + 11) + (57 + 10 + 28)
]
Calculating each class:
- Freshman: (52 + 26 + 21 = 99)
- Sophomore: (52 + 20 + 22 = 94)
- Junior: (60 + 20 + 11 = 91)
- Senior: (57 + 10 + 28 = 95)
Now summing these totals:
[
\text{Total} = 99 + 94 + 91 + 95 = 379
]
Part (a): Probability that a participant is satisfied
To find the probability that a randomly selected participant is satisfied, we sum the number of satisfied students:
[
\text{Satisfied} = 52 + 52 + 60 + 57 = 221
]
Now, we calculate the probability:
[
P(\text{satisfied}) = \frac{\text{Number of satisfied students}}{\text{Total number of students}} = \frac{221}{379}
]
Calculating this gives:
[
P(\text{satisfied}) \approx 0.583
]
Rounded to three decimal places:
[
P(\text{satisfied}) \approx 0.583
]
Part (b): Probability that a participant is a junior
To find the probability that a randomly selected participant is a junior, we need the total number of juniors:
[
\text{Juniors} = 60 + 20 + 11 = 91
]
Now, we calculate the probability:
[
P(\text{junior}) = \frac{\text{Number of juniors}}{\text{Total number of students}} = \frac{91}{379}
]
Calculating this gives:
[
P(\text{junior}) \approx 0.240
]
Rounded to three decimal places:
[
P(\text{junior}) \approx 0.240
]
Part (d): Probability that a participant is satisfied or a junior
To find (P(\text{satisfied or junior})), we use the formula for the union of two events:
[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
]
Where:
- (A) is the event of being satisfied.
- (B) is the event of being a junior.
We already have:
- (P(A) = P(\text{satisfied}) \approx 0.583)
- (P(B) = P(\text{junior}) \approx 0.240)
Now we need (P(A \cap B)), which is the probability of being both satisfied and a junior. The number of satisfied juniors is 60.
[
P(A \cap B) = \frac{60}{379}
]
Calculating this gives:
[
P(A \cap B) \approx 0.158
]
Now we can calculate (P(A \cup B)):
[
P(\text{satisfied or junior}) = P(\text{satisfied}) + P(\text{junior}) - P(\text{satisfied and junior})
]
Substituting the values:
[
P(\text{satisfied or junior}) \approx 0.583 + 0.240 - 0.158
]
Calculating this gives:
[
P(\text{satisfied or junior}) \approx 0.665
]
Rounded to three decimal places:
[
P(\text{satisfied or junior}) \approx 0.665
]
Summary of Results:
- (a) (P(\text{satisfied}) \approx 0.583)
- (b) (P(\text{junior}) \approx 0.240)
- (d) (P(\text{satisfied or junior}) \approx 0.665)