Probability of Pregnancy I have a probability question, I am not sure how to make of it.   Hope you can give me some guidance, math has never been my strong subject.
Assuming the lady is 36 years old and is considered part of the "black" line which is 10% chance of pregnancy per cycle, and the lady gets a 6 cycle for that year, what is the probability for pregnancy for that year?

My answer is .469?

And if possible, can you point out the name of the concept behind the equation so I can check it out in youtube.  Thanks a lot.
 A: 
If possible, can you point out the name of the concept behind the
equation

The equation that you have been employing in your table relates to the geometric distribution and this relates to 'waiting time', a concept in queueing theory (see overviews of this topic on math and stats sites of stackexchange).
$$P(\text{success in the $k$-th cycle)} = (1-p)^{1-k}p$$
for $p_{succes} = 0.1$ you get
$$\begin{array}{| r | cccccccc|}
\hline
k & 1 & 2 & 3 & 4 & ... & n\\ \hline 
\text{no success after $k$ attempts}& 0.9 & 0.81 & 0.73 & 0.66 & ... & (1-p)^k \\
\text{success in $k$-th attempt}& 0.1 & 0.09 & 0.08 & 0.07 & ... & p(1-p)^{k-1} \\
\text{cumulative}& 0.1 & 0.19 & 0.27 & 0.34 & ... & 1- (1-p)^{k} \\
\hline
 \end{array}$$
Some intuitive display of the equations:
$$\begin{array}{} P(\text{success at $k$-th cycle)} &=& \underbrace{P(\text{no success $ \leq k-1$)}  }_{(1-p)^{k-1}}\times \underbrace{P(\text{success in a single cycle})}_{p}\\
P(\text{success $ \leq k$)} &=& 1- \underbrace{P(\text{no success $ \leq k$)}}_{(1-p)^k} 
\end{array}$$

Of course, this is only an idealized model. As mentioned in the comments, the probability for becoming pregnant (success) is not constant from cycle to cycle. Also, you do not really know this probability for a single person.
The statistics that you often read about might be measurements of pregnancy probabilities over longer periods (and do not really determine directly the cycle probabilities). These are only expressed in terms of probability per cycle, but that is not how the random process works. It is wrong to consider groups of couples that are trying to get pregnant as a homogeneous group in which each couple has each cycle equal probability of getting pregnant.
For instance, consider the opposite situation: couples trying to avoid pregnancy.

*

*For this situation you often read about those statistics as the probability that you may still get pregnant when using some type of contraception (say, when using a condom there is still a 2% probability of getting pregnant each time, or 15% each year). However, this probability is a lot dependent on the type (and frequency) of use.
The stochastic process of becoming pregnant, the cause of distribution in pregnancy, should not be interpreted as a process in which each couple has this similar x percent probability each sex/cycle time. It is much more like some couples will use the condoms badly and have a high probability of getting pregnant each time, and others will have a proper use of the condoms (and many other factors play a role as well besides just 'proper' use) and have a much lower probability. It is not like the 15% of people that get pregnant each year while using condoms, where just unlucky (or lucky from some alternative point of view) to hit one of those 2% of cases. It is much more likely that you get a 15% group that has high probability of getting pregnant each cycle and a 85% group of people that has a low probability of getting pregnant each cycle.
How is this important? So, the curves that you read (like the black orange and green lines in your graph) are used for representation of probabilities of getting pregnant each cycle. But it would be wrong to interpret this as a simple geometric distribution and believing that the probability to get pregnant is time dependent with the factor according to that graph (like you can still have a high probability as long as you try hard enough, even when you fall in a group with statistical low probability). Couples will have variation in their probability to get pregnant. This needs to be taken into account.
How does this work out? Say you are using some statistic that measured that the probabilities of 36 yr old women (with 13 cycles per year) getting pregnant during a year was 75% then this may get converted to the 10% pregnancy probability each cycle. But this is for representation. It is not truly like that. It is very likely that it is some people having lower pregnancy probability and others having higher probability rates. It could be three quarters having a high 20% probability each cycle and a quarter of the couples having a low 2% probability each cycle (or something more continuous). This situation would look like the image below. Both situations show a 75% of the people pregnant after one year (13 cycles), but there is a strong difference over longer periods. A quarter of the couples would have a very hard time getting pregnant, which is not suggested by the geometric distribution with a single 10% per cycle probability (in which case the curve rises more strongly when having more than 13 cycles).

The bottom-line is that you need to critically review your sources for these 'per cycle' probabilities and consider what the raw data originally meant to express. And also you need to consider that these probabilities are very crude/rough estimates that are far from the rigorous impression that you might get from the equations and "scientific" expressions.
(For your family-member, who has fewer menstrual cycles, this might be good news, since this viewpoint shows that the reported probabilities of pregnancy are less dependent on simply trying a lot of cycles. Having less or longer cycles does not, so much, reduce probabilities of getting pregnant. Except, of course, when the lower number of menstrual cycles relate to a wider set of fertility related problems.)
A: 0.469 is the right answer. The Math is quite simple if we "cheat" a little bit and simplify the problem by calculating the probability of NOT getting pregnant instead:
On each cycle, there is a 90% chance of getting no pregnancy. If we are on the remaining 10%, we're done. In other case, on the next cycle, there will be another 90% of non-pregnancy chance (with 90% of 90% being 81%). Then, after three cycles, we are left with 90% of 81%, or 72,9%
In conclusion, at the $n$th, the non-pregnancy chance will be $0.9^n$ (times 100 if you want a percentage) For example, after six cycles, 0.9^6 is about 0.531, so 53.1% faillure chance (or, if you prefer, 46,9% success chance)
For more info on why this holds, take a look at conditional probalbilities!
