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[quote=Anonymous]From reddit: http://www.reddit.com/r/statistics/comments/1pbjlf/can_someone_help_me_use_probability_to_make_this/

You're on the right track, TysonStoleMyPanties, but it turns out that the problem is a little more complicated.
Given that the trials are independent (big assumption), the probability of success in each trial is 0.4. So the expected number of trials until success is 1/0.4 = 2.5. We could say that 2.5 trials would cost $30k, at $12k per trial, if we could pay for "half" or "partial" trials; however, I doubt that paying for partial trials is a possibility. We could pay for one, two, three, and so on trials.
Using the geometric distribution, we can calculate the probability of success p in any given number of trials k, where the probability density function is given by
pdf = p*(1-p)k-1.
We see that the probability of success within the first two trials is the sum of the probability of success in the first trial (pdf=0.4) and the probability of success in the second trial (pdf=0.24), which is equal to 0.64 -- meaning that there is a 64% chance of success within the first two trials. From this, we also know that the probability of success in three or more trials is 1-0.64 = 0.36.
We can calculate the pdf for each possible number of trials, but for the sake of convenience, I calculated the pdf for up to 12 trials. Here is the graph showing the calculated pdf for each of the first twelve trials along with the corresponding cost at $12k/trial.
In order to choose the more cost-effective strategy, we could compare the expected cost of the pay-per-trial option with that of the unlimited trials option. The unlimited option costs $25k, which is slightly more than the cost of two pay-per-trial options ($24k), but considerably less than the cost of three pay-per-trial options ($36k). The more trials it takes, obviously, the more cost-effective the unlimited option would seem.
At this point, it may seem intuitive to select the unlimited option because one would save a considerable amount of money if three or more trials were attempted; however, don't forget that one has a 64% chance of success within the first two trials.
The strategy I would recommend depends upon how many unsuccessful trials the OP would be willing to attempt given the unlimited option had been purchased. Realistically, after a certain number of failed attempts, the OP's wife will quit attempting to get pregnant using this method of IVF, and the $25k will be a sunk cost.
Let's say that the OP's wife would quit after five attempts. If they were using the pay-per-trial option, then they would have spent $35k ( =$60k - $25k ) more than had they used the unlimited option. But don't forget that there is only a 5.18% chance of that happening. Whereas, there is a 40% chance that the pay-per-trial option would save them $13k (i.e., there is a 40% chance of success within the first trial), and a 24$ chance that the pay-per-trial option would same them $1k. So, to compare the expected costs, we simply sum the difference in cost for each possibility from one to five trials (assuming the OP would stop after five unsuccessful trials), weighted by their respective probabilities.
For example: Suppose that the outcome of each trial is independent of the previous outcomes. Also suppose that the probability of success is 0.4. Further suppose that the OP's wife would cease the series of IVF treatments after five unsuccessful trials, regardless of the payment scheme. The expected money saved using the pay-per-trial option compared with the unlimited option is given by
0.4($25k - $12k) + 0.24($25k - $24k) + 0.144($25k - $36k) + 0.0864($25k - $48k) + 0.05184 ($25k - $60k)
= $0.0544k
= $54.40.
In conclusion, given the above assumptions, using the pay-per-trial option is expected to be slightly more cost-effective than the unlimited option. Please note that this estimation is sensitive to several factors. For instance, we have not incorporated risk aversion into this model. That is, if the OP's wife cannot afford paying ten's of thousands of dollars then purchasing the unlimited option may be the best option for her. Also, if the OP's wife would be willing to attempt more than five unsuccessful trials, then the unlimited option would be preferable. Finally, if the probability of success is less than 40%, but everything else remains unchanged, then the unlimited option may be the best way to go.
OP, I'd be happy to crunch some numbers or explain the details if you would like more help estimating the cost-effectiveness given a different set of assumption.[/quote]

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