On average 62 % of Finite Mathematics students spend some time in the Mathematics Department's resource room. Half of these students spend more than 90 minutes per week in the resource room. At the end of the semester the students in the class were asked how many minutes per week they spent in the resource room and whether they passed or failed. The passing rates are summarized in the following table:
Time spent in resource room Pass %
None 27
Between 1 and 90 minutes 48
More than 90 minutes 81
If a randomly chosen student did not pass the course, what is the probability that he or she did not study in the resource room?
Why did the failed student avoid the resource room?
Because they heard it was a room full of resources, but they were looking for a room full of distractions instead!
Sorry, back to your question. Let's calculate the probability!
The probability that a randomly chosen student did not study in the resource room can be found by considering the students who did not pass the course and did not study in the resource room. We will divide the number of students who did not pass and did not study in the resource room by the total number of students who did not pass.
From the given information, we know that:
- Students who did not pass and did not study in the resource room: 27%
- Students who did not pass and studied between 1 and 90 minutes in the resource room: 48%
- Students who did not pass and studied more than 90 minutes in the resource room: 81%
To find the probability that a randomly chosen student did not pass and did not study in the resource room, we add up the percentages:
27% + 48% + 81% = 156%
Wait, that's not right! It seems like there's a mistake in the information provided, as the percentages add up to more than 100%. This can happen sometimes when dealing with statistics, where there might be some overlapping categories or errors in the data.
Without accurate percentages, it's not possible to calculate the probability accurately. We need to know the correct values for each category before determining the probability that a failed student did not study in the resource room.
To find the probability that a randomly chosen student who did not pass the course did not study in the resource room, we need to calculate the probability that the student falls into the "None" category of time spent in the resource room.
From the given information, we know that the passing rates for each category of time spent in the resource room are as follows:
None: 27%
Between 1 and 90 minutes: 48%
More than 90 minutes: 81%
To find the probability that a student did not study in the resource room given that they did not pass, we need to consider the total number of students who did not pass and did not study in the resource room (None category), divided by the total number of students who did not pass.
Let's denote the probability of not studying in the resource room and not passing as P(Not Study | Not Pass). Mathematically, it can be calculated as:
P(Not Study | Not Pass) = Number of students not studying in the resource room and not passing / Number of students not passing
From the given data, we know that the passing rates are as follows:
P(None and Not Pass) = 27%
P(Between 1 and 90 minutes and Not Pass) = 48%
P(More than 90 minutes and Not Pass) = 100% - Passing rate
We need to determine the value of the expression P(None and Not Pass).
The total passing rate is given by:
Total Passing Rate = P(None) * P(Not Pass | None) + P(Between 1 and 90 minutes) * P(Not Pass | Between 1 and 90 minutes) + P(More than 90 minutes) * P(Not Pass | More than 90 minutes)
Since we know the passing rates for each category, we can calculate the value of P(None and Not Pass) as follows:
P(None and Not Pass) = P(None) * P(Not Pass | None) / Total Passing Rate
Plug in the values:
P(None and Not Pass) = (27% * 1) / (27% + 48% + (100% - Passing rate))
Simplifying,
P(None and Not Pass) = 27% / (27% + 48% + (100% - Passing rate))
Now, to find the probability that a randomly chosen student who did not pass the course did not study in the resource room, we can use the formula for conditional probability:
P(Not Study | Not Pass) = P(None and Not Pass) / P(Not Pass)
Using the values we have,
P(Not Study | Not Pass) = P(None and Not Pass) / (27% + 48% + (100% - Passing rate))
This will give us the probability that a randomly chosen student who did not pass the course did not study in the resource room.
To find the probability that a randomly chosen student did not study in the resource room given that they did not pass the course, we need to use conditional probability.
First, let's calculate the probability of a student not passing the course. This can be done by summing up the pass percentages for each time category:
Pass % = None + Between 1 and 90 minutes + More than 90 minutes = 27 + 48 + 81 = 156
Next, we need to find the probability of a student not passing the course and not studying in the resource room. From the given information, we can see that the "None" category corresponds to not studying in the resource room. So we can simply use the given pass percentage for the "None" category:
Probability of not passing the course and not studying in the resource room = None Pass % = 27
Finally, we can calculate the probability that a randomly chosen student did not study in the resource room given that they did not pass the course:
Probability of not studying in the resource room (given not passing) = (Probability of not passing the course and not studying in the resource room) / (Probability of not passing the course)
Probability of not studying in the resource room (given not passing) = 27 / 156 ≈ 0.173
Therefore, the probability that a randomly chosen student did not study in the resource room given that they did not pass the course is approximately 0.173 or 17.3%.