# Two sequences for a chance at reward. Which should I choose?

this is my first Cross Validated post!

I am trying to figure out how to choose between two different sequences to obtain a desired outcome. This is the best way I can think of to desribe the situation:

In option 1 there are two events. In the first event there is a 5% chance to obtain the reward. In the second event there is a 9% chance to obtain the reward. You can get the reward from each event.

In option 2, there are two events. In the first event there is a 0% chance of the reward. In the second event there is a 12% chance of getting the reward.

Which sequence has the highest total probability of getting the reward and why?

I am a complete neophyte when it comes to statistics, so I am hoping to start learning here! Thank you for your help!

Edit: In each Option, the sequence of event A and event B always happen. Each take approximately 5 minutes each. At the end of each event, regardless of the Option, a random number generator generates a "reward"; the probability that it is the desired "reward" are those stated above.

Assuming we get past these communication issues, I would be curious how difference in the length of each option would affect the outcome.

Are there other data needed? Thank you!

Edit 2: Ok, I've tried to capture this in a picture, because that's how I'm thinking about the situation and clearly my attempts to write it out have not been successful.

Each rectangular block is what I've been referring to as an "event". Every event, once you are on a path happens 100% of the time; their occurrence is an intrinsic part of the option.

I am assuming that with Option B, the chance of getting the reward really is just 12.2% of every time I select Option B. For Option A, with the two options, I do not understand how to combine them together into a single "number" that represents that Option's potential to result in 1 (or possibly 2) rewards, so that I can compare it against taking Option B. I am assuming that there is a statistical method that will allow me to make an apples to apples comparison of the two Options so that I can understand which would be better to take (where better = highest likelihood of reward).

Is that any better?

• You don't supply enough information. You also need to stipulate how often each event occurs.
– whuber
Commented Aug 17, 2017 at 18:46
• Your edit still doesn't supply the requisite information. What can you tell us about the relative frequencies of events $A$ and $B$? Are you trying to say that event $A$ happens once, followed by event $B$, for a total of 10 minutes? If so, then please state that explicitly--and then explain what you mean by the "length of each option."
– whuber
Commented Aug 17, 2017 at 21:25
• First case probability of winning, which is 1-(probability of not winning) is 1 - .95×.91=0.1355 and in second case 1 -1×.88=0.12, so first case is better odds of an award?
– user137329
Commented Aug 18, 2017 at 1:30

For Option A your chance of getting both rewards is $0.059\times 0.086 = 0.5074\%$. To obtain exactly one reward, this can be done by succeeding at the first event and failing the second, or failing the first event and succeeding at the second. That is, your chance is $0.059\times(1-0.086)+0.086\times(1-0.059)=13.4852\%$. And your chance of getting at least one prize is the sum of the two, or a smidgeon under 14%.