Making an inference from conditional probability values

Consider the following data about bulb failures (which require a filament replacement for repair) compared across three companies:

          P(1st failure) | P(2nd failure | 1st failure) | P(3rd failure | 2nd failure)
--------------------------------------------------------------------------------------
Company A    1 in 4.0    |          1 in 5.9            |          1 in 6.1
Company B    1 in 3.1    |          1 in 6.1            |          1 in 9.2
Company C    1 in 18     |          1 in 8.1            |          1 in 4.0


This data was subsequently used to plot the following graph:

The values were obtained by just dividing the numbers in table (e.g., 1/4.0, 1/5.9, 1/6.1 to obtain the bars for Company A and so on). From hereon, the following inference was made:

• The probability of first failure is high for Company A and Company B but Company C bulbs are robust
• After the first failure, there is a decreasing trend for Company A and Company B indicating that their filament replacements are effective. However, this is not the case for Company C.
• After the second failure, the increasing trend for Company C is even more visible showing that filament replacements by Company C are not effective. Therefore, it makes sense to consider replacing bulbs from Company C when they fail (assuming the cost of repair is very high for each failure).

I have two specific questions:

• Does these inferences make sense? If yes, the graph plots probability and conditional probability bars next to each other. Is that correct or are there better ways to show this?
• If the inferences themselves are incorrect, what should I infer from the given data? And what is the best way to show this in a graph?

1 Answer

The "correct" inference depends on what problem you're trying to solve ;) .

That said, it's pretty clear this is about buying bulbs, so...

One thing I think you're missing from your analysis is putting numbers to all these things. For example, consider two replacement policies:

• Buy all bulbs from company C. When they fail, repair the filament.
• Buy all bulbs from company C. When they fail, replace the bulb.

You can work out the cost of these two policies in terms of the cost of a new bulb, and the cost of a filament. From there, you can then specify exactly how much more expensive a bulb replacement must be than a filament replacement (in terms of the cost of a single bulb) for one policy to be better than another. That tells you not just what buying strategy you shoudl follow right now, but what changes in relati ve price you need to look out for to decide when that strategy would change.

Similarly, you can compare buying bulbs from company A vs company C, and put an exact number on many times more expensive than the former the latter must be before buying from A is most cost effective than buying from C.

In short, the sort of inferences you are drawing look ok, but can be made more exact.

• Thank you for your response. I do agree that cost should be factored in when looking at this. I will look into this in more detail. Can you also answer my other fundamental question of comparing a probability with conditional probability? I want to understand if this is a fair comparison because the population is different. Commented Mar 12, 2014 at 5:31
• I'm not aware of any standard conventions on this. My opinion is that the way you've presented it is fine. I clearly understood what was being shown. Also, conditional probability highlights the difference in filament replacement trend between company C vs A or B more readily any reasonable alternative method I can think of. But, as said, that's all just my opinion - I don't have anything to back it up with beyond gut feeling. If anyone else answers with a more concrete justification you should probably listen to them.
– Pat
Commented Mar 12, 2014 at 8:43