# negative binomial / wording issues

Q> 60% of the buses at my local bus stop are on route 39. If the routes of different buses are independent of each other, what is the probability that at least 4 buses that are not route 39 arrive before the 3rd route 39 bus ?

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I really don't understand these types of questions. The wording throws me for a loop I know that it is probably neg binom but even with the formula I draw a blank for how to organize everything.

'at least' ? - is this the reason that this problem can't be done with only 1 application of the neg binom formula ?
'arrive before' ? - what if it were 'arrive on' or 'arrive after' ? how would these effect the total of successes+failures 'n' ?

#### chiro

MHF Helper

The best advice I can give for you is to think about what each distribution is modelling process wise and not mathematically/probabilistically/statistically.

For example binomial represents a physical model where you have n trials that are independent from each other with either a 0 or 1 outcome and then we sum up the outcomes to get X. Some say coin tosses or other things, but the idea is that you have n independent things with a success/fail for each and then look at the distribution for getting so many successes.

Now negative binomial processes model a process that waits for so many failures to occur. The geometric process is a special case waiting for one failure to occur but negative binomial allows arbitrary numbers of failures.

If you try and think in formulas only then I can see why this stuff would be confusing: the math needs to be supplemented by ideas with some kind of physical, or visual intuition and if your teacher hasn't communicated this then they aren't really a good teacher.

Hi thanks for the general ideas but I was looking for something a little more specific in regards to understanding the negative binomial formula. I'm not sure how to handle the variety of ways these questions can be asked in a general way. For instance, is this one going from (8-1 choose 4-1) successes ? What if it were worded differently like I asked in OP ?

#### chiro

MHF Helper
It's a little hard to answer your question, but basically you are looking for the relation to the general characteristics of the process.

Negative binomial models the process where after x trials you get k failures: the first k-1 failures can happen anywhere in-between but the last failure always happens at the xth trial (we also need k > 0 and k be a whole number).

The thing is though that if you go further and further, you will have to derive your own distributions and this is not easy (remember that someone actually had to create all this stuff from scratch or at least build on all the results that have been done off the labor of other people before them).

So yeah the short answer is look at the process (what is the condition of the distribution, how do these assumptions relate to the process, are they exact, close enough to use) and then relate that to the best model possible.