probability

[PG1995]

Hi

Could you please tell me what this "red card" is? Thanks.

Which of the following are pairs of mutually exclusive events in the drawing of a single card from a standard pack of 52 cards
(a) An ace and an even number
(b) A heart and a queen
(c) A club and a red card
(d) An even number and a spade

Regards
PG

 

[steveB]

A red card is any card from the suits of diamonds and hearts. Typically, those cards use red fonts and pictures, while clubs and spades use black fonts and pictures.

 

[jpanhalt]

With that information, do you need any help with the answer? Hint, two of the situations are mutually exclusive.

John

 

[PG1995]

Thank you, Steve, John.

@John: Thanks for the input. I was only confused about the "red card".

Regards
PG

 

[PG1995]

Hi

Could you please help me with this query(ies)? Thank you.

Regards
PG

 

[steveB]

It seems to me that you did it correctly.

What you can plot is probability density versus number of patients. Obviously the number of patients is a discrete variable so you have p(n)=f(n), where n is the number of patients and p(n) is the probaility of getting n patients on any given day.

If you don't yet have a probability density, and only have data points. Then you can plot the number of days that you see a number n, versus n. In other words, let N(n) represent the number of days N that you saw n patients, and then plot N(n) versus n. Eventually, if you get enough data, you can estimate the probalility density as p(n)=N(n)/sum(N), which just says that the probability of observing n patients on a given day is equal to the number of times you previously saw n patients divided by the number of days you saw patients.

In your case you have very limited data. Still, for seriously ill patients you can plot N(n)=1, if n=33, 50, 22, 27, 48, n=0 otherwise; and, for routine patients you can plot N(n)=1, if n=34, 31, 37, 36, 27, n=0 otherwise.

 

[PG1995]

Thank you, Steve.

If you don't yet have a probability density, and only have data points. Then you can plot the number of days that you see a number n, versus n. In other words, let N(n) represent the number of days N that you saw n patients, and then plot N(n) versus n. Eventually, if you get enough data, you can estimate the probalility density as p(n)=N(n)/sum(N), which just says that the probability of observing n patients on a given day is equal to the number of times you previously saw n patients divided by the number of days you saw patients.

Even if I have done it correctly, I still don't have clear picture of it because I can't picture its graph mentally. In most bell shaped diagrams frequency is drawn on y-axis and x-axis is used for the data. But in this case I think I need to plot the number of patients on x-axis but then what will I plot on y-axis? Besides this number of patients, to me, sounds more like frequency than data. Please guide me. Thanks.

Regards
PG

 

[steveB]

I still don't have clear picture of it because I can't picture its graph mentally. In most bell shaped diagrams frequency is drawn on y-axis and x-axis is used for the data. But in this case I think I need to plot the number of patients on x-axis but then what will I plot on y-axis? Besides this number of patients, to me, sounds more like frequency than data. Please guide me. Thanks.

Did you understand my post above? I tried to explain that the frequency is on the y-axis and the data is on the x axis. The data is the number of patients. The frequency is the "number of days" that meet the criteria for the data.

Let's take another example. What if you had the following data?

# Patients ...... # Days
--------------------------
0 ..................... 1
1 ..................... 0
2 ..................... 3
3 ..................... 2
4 ..................... 4
5 ..................... 8
6 .................... 15
7 .................... 12
8 ..................... 6
9 ..................... 3
10 ................... 2

Now plot this data with # patients on x-axis and number of days on the y-axis. You would see a bell-shaped distribution for the 56 days you took data for. If you then plot number of days divided by 56, versus number of patients, you will have a probability distribution.

 

[jpanhalt]

There is not just one way to plot those data. One common method is the Levey-Jennings chart. James Westgard expanded on that charting method to develop action points.

Basically, the number of patients is plotted on the Y-axis. Sequential days on the X-axis. The mean and multiples of the SD are highlighted on the Y-axis.

Here are some links to Westgard and Levey-Jennings (second link):

https://www.westgard.com/westgard-rules-and-multirules.htm
https://en.wikipedia.org/wiki/Laboratory_quality_control

You can apply any theorem you want for the bounds. Westgard is focused on parametric data that are normally distributed; hence, his use of 2 and 3 SD.

John

 

[PG1995]

Thank you.

I still don't get it. This is how I see it if I plot number of patients on x-axis and it makes no sense to me. Now please help me with it.

Regards
PG

 

[steveB]

The y axis is not a time axis, it is a "frequency or occurrence" axis. It is the number of days that you see n patients. It doesn't matter exactly which day you see 4 patients, but if you see 4 patients on 12 different days then y(4)=N(4)=12. The variable n takes on all integer values between 0 and 6 billion. For every value of n, there is a particular data value N(n) that represents the number of days that the doctor saw n patients.

Please reread all of my posts that relate to answering this question. Read them very carefully, sentence by sentence. The answer is there. I've placed it there is a couple of different ways. Read carefully and try to understand exactly with the N(n) and p(n) are, as I described them. Whatever concept you have in your mind now, throw it away. It is wrong. Think carefully about what the right way is.

 

[jpanhalt]

What variables you plot where depends on what you want to show and, most important, why you are collecting and plotting the data.

If it is an assignment, and you need to show the frequency distribution to demonstrate that the results look like a normal distribution, then steveB has shown you how to do that with his hypothetical data set. However, the data in this case are insufficient to show such a nice curve.

If I were Dr. Levinson and needed to manage my practice, I would be more concerned about how to use the data to improve my practice's efficiency. He has already assumed certain characteristics of the two patient groups. I do not read the question as asking you to confirm those assumptions nor are they particularly important to the solution.

That is why I would use a Levey-Jennings approach. I would want to know that my office was appropriately, but not over staffed. Dr. Levinson also needs to determine whether any degree of understaffing is acceptable.

Moreover, in the real world where there are different third-party payers (such as in the USA) and other variables affecting patient visits, I would probably use different color dots or pins to indicate the different major payers or other variable and plot the number of patients per day.* The question doesn't say how many days a week Dr. Levinson's office is open. Assuming it is open 5 days a week, that is a pretty low workload for an outpatient practice. I would want to know why my daily visits varied by such a large amount (i.e., average about 10 to 16).

John

*Staffing levels are usually based on days or even hours, not weeks. For example, it is more common to have someone work only on Tuesdays and Thursdays, rather second and fourth weeks of a month.

 

[steveB]

PG,

One thing about my description that may have confused you is that I did my example by "day" rather than by "week" as in your example. But the concept is the same no matter which units you choose. As jpanhalt mentioned (and I did also), the data in your example is so sparse that you can't really see a normal distribution in the way you are thinking. This is probably making it harder for you to visualize the usual way you like to plot the data. I agree with jpanhalt that there are other ways to display the data, but I am taking my best guess at the way you are used to plotting data and I am trying to help you visualize the concept of what "entity" relates to "frequency of occurrence". In this case the number of weeks N that a certain number of patients n is observed is the variable that represents the frequency of occurrence, and N can be used to generate a probability distribution if enough data is collected. Thinking this way should help you visualize to "have clear picture" and "picture its graph mentally" as you wanted to do. Once you succeed in doing that, consider japanhalt's input as well because answering the question and visualizing the scenario is important, but what is more interesting and important (in the long term) is being able to solve real problems, such as staffing an office.

Steve

 

[PG1995]

average inventory (frequency distribution)

Hi

Could you please help me with this query? Thank you.

Regards
PG