StatB

I apologize but I would actually reiterate the same saying already mentioned by @Stat Assistant Professor that it becomes somewhat difficult for me to directly gauge your chances at math departments. You should follow his advice of digging more in mathematics forum or mathematicsgre website. The main hindrance in giving feedback on the math departments is that I am not that much knowledgeable about how the admission and selection process goes on these departments. I know some of our seniors who received  both their undergraduate and master's degree at ISI (in Statistics) and went to do probability research in math departments (like Columbia). But that number is small and many from the Kolkata center normally prefer joining Statistics departments (though they have solid mathematics training at undergraduate level, the course curriculum is more skewed to Statistics. Hence they might feel more comfortable at Stat departments) In fact as I have already mentioned, many of my batchmates who did BMath and MMath from ISI Bangalore, and were very much keen to do stuffs related to probability, have chosen Statistics departments for doing their PhDs rather than mathematics departments. Thus, I am pretty sure that you will have good to decent chances in the Stat departments already mentioned in the earlier answers. Regarding Chicago Stats program, it considered as one of the top programs often selecting mostly among the top students from MStat (This time they have selected one from MMath too). So if you do extremely well in your three semesters of MStat, get strong letters then you might try applying there and have a shot. 
Somehow I feel that, your undergraduate degree from CMI would be a plus point if you want to apply to math departments. I know that many top students from CMI are able to get PhD position in pure mathematics (too in universities like Chicago) right after their BSc. So, you might have good chances in math departments, but I will not be able to make a detailed statement on it. 
There are many good European universities where you can apply. As an alumni from CMI (plus that your are presently at ISI) you have very good chance in ending up in École normale supérieure (Paris). They have some connection with CMI and prefer CMI students in their lab. In that case, you would require someone from CMI who can write a strong letter on behalf of you. Some other universities which you can look are :- université Paris-Dauphine, PSL, Paris Saclay, Paris-Sud and so on. (You might look mathematicsgre.com or reddit to know more about them). There are couple of good universities in UK where predominantly probability research is carried out. Apart from oxbridge, Bath and Warwick comes to my mind. You can have a look at http://people.maths.ox.ac.uk/tanner/Prospects2010/JNorrisTalk.pdf to know more about probability dominated departments (But I guess the document is old, so you might search whether the peoples mentioned in the documents are still supervising students or not.) Another destination in Europe which comes to mind when talked about probability and random graphs is Netherlands. I have already mentioned about NETWORKS program which have selected a couple of ISI students recently. Apart from that we know some of our seniors who graduated from universities like Eindhoven, Leiden etc., and have done groundbreaking works in applied probability.
 

I apologize but I would actually reiterate the same saying already mentioned by @Stat Assistant Professor that it becomes somewhat difficult for me to directly gauge your chances at math departments. You should follow his advice of digging more in mathematics forum or mathematicsgre website. The main hindrance in giving feedback on the math departments is that I am not that much knowledgeable about how the admission and selection process goes on these departments. I know some of our seniors who received  both their undergraduate and master's degree at ISI (in Statistics) and went to do probability research in math departments (like Columbia). But that number is small and many from the Kolkata center normally prefer joining Statistics departments (though they have solid mathematics training at undergraduate level, the course curriculum is more skewed to Statistics. Hence they might feel more comfortable at Stat departments) In fact as I have already mentioned, many of my batchmates who did BMath and MMath from ISI Bangalore, and were very much keen to do stuffs related to probability, have chosen Statistics departments for doing their PhDs rather than mathematics departments. Thus, I am pretty sure that you will have good to decent chances in the Stat departments already mentioned in the earlier answers. Regarding Chicago Stats program, it considered as one of the top programs often selecting mostly among the top students from MStat (This time they have selected one from MMath too). So if you do extremely well in your three semesters of MStat, get strong letters then you might try applying there and have a shot. 
Somehow I feel that, your undergraduate degree from CMI would be a plus point if you want to apply to math departments. I know that many top students from CMI are able to get PhD position in pure mathematics (too in universities like Chicago) right after their BSc. So, you might have good chances in math departments, but I will not be able to make a detailed statement on it. 
There are many good European universities where you can apply. As an alumni from CMI (plus that your are presently at ISI) you have very good chance in ending up in École normale supérieure (Paris). They have some connection with CMI and prefer CMI students in their lab. In that case, you would require someone from CMI who can write a strong letter on behalf of you. Some other universities which you can look are :- université Paris-Dauphine, PSL, Paris Saclay, Paris-Sud and so on. (You might look mathematicsgre.com or reddit to know more about them). There are couple of good universities in UK where predominantly probability research is carried out. Apart from oxbridge, Bath and Warwick comes to my mind. You can have a look at http://people.maths.ox.ac.uk/tanner/Prospects2010/JNorrisTalk.pdf to know more about probability dominated departments (But I guess the document is old, so you might search whether the peoples mentioned in the documents are still supervising students or not.) Another destination in Europe which comes to mind when talked about probability and random graphs is Netherlands. I have already mentioned about NETWORKS program which have selected a couple of ISI students recently. Apart from that we know some of our seniors who graduated from universities like Eindhoven, Leiden etc., and have done groundbreaking works in applied probability.
 

Hi @aldoraine. Apologies for late reply. It will be also better if you can express the name of your undergraduate institution. If you have completed BMath from ISI Bangalore or BSc Mathematics & Physics at CMI, then your profile seems to be a solid one. I guess a BS from IIT Kanpur or Bombay would also have more or less similar pedigree, but I would rank them low as compared to ISI and CMI as far as undergraduate mathematics is concerned. (My answers will be based on the assumption that your undergrad was from one of these institutes.) You have good percentage in your first semester of MStat. I would ask you to keep the same progress. You definitely have a good chance of ending up in the schools already mentioned by @Stat Assistant Professor. One of our seniors who had similar profile like you and was keen on doing work in probability (undergrad from ISI, slightly more percentage in MStat than you.) ended up getting offers from Brown university for PhD Mathematics. He has ultimately joined NETWORKS program at Netherlands (You can also look at this as I have recently heard that NETWORKS is getting more popular. Prof. Rajat Subhra Hazra is also at Leiden on sabbatical.) 
Some of our batchmates who did undergrad and masters both in ISI Bangalore and were wanting to do Statistics/Probability has received offers from Cornel, Chicago, UT Austin, ETH Zurich (Master's ), UNC Chapel Hill etc. (All are Statistics programs). The students getting offers from Cornel, Chicago, Chapel Hill were ranked very high in their classes, had a extra projects and also a very good connection with professors which resulted in strong recommendations. So, it will be also good if you can try interacting with some of your professors who will be ultimately writing recommendation for you. Under the assumption that you had undergrad from CMI/ISI, your profile will be similar to the students mentioned above. Particularly, if you wish to do probability, you will encounter more probability courses in MStat than any other course at ISI (provided you take the combinations in that way and also provided the intended courses are being offered. Caution:- Not all the courses in probability specialization are offered every year). So these schools may be your "reach". Apart from that you can apply to any other universities at the level mentioned in the earlier answers. I can see them as very positive for you. 
Regarding extra projects, I guess it will not be a big problem. Try working on your project (assuming that you will take "Project" as one semester course in the final semester) from third semester itself so that you can mention about it in your SoP as well as CV.  But note one thing:- most of the advanced courses in probability specialization (like weak-convergence and empirical processes, martingale theory, brownian motions, large deviations, percolation) will be given in last year. SO, you may not be able to show evidence of these courses (or grades) while applying. But you can definitely write in your SoP. Some universities, though, ask grade-sheets of third semester around January.

You have a strong pedigree, so your application should definitely be 'in the discussion' at a lot of schools. Your undergrad performance looks very strong, and your Masters performance at ISI looks quite strong as well. I have seen students with scores of high 80's from ISI graduate "with distinction" and get accepted to top PhD programs in the U.S.A., so it seems like anything over an 80 is considered very good from ISI.  
I think you should have a shot at schools at least at the level of University of Minnesota and University of Wisconsin (in particular, you are interested in Markov chain Monte Carlo theory -- which tends to be a bit more probability theory-focused than statistics, then there are some good faculty at UMN for this... UWisc also have Timo Seppalainen working on probability theory who can supervise Stats PhD students). The STOR program at University of North Carolina-Chapel Hill also has some good probability theory/stochastic processes faculty. I would definitely recommend applying to UMN, UWisc, and UNC-CH. A lower ranked Statistics program that has some great probability faculty is Michigan State University. You could also apply to UC Berkeley as a (far) "reach" school. 
In general, though, Statistics PhD programs outside of top programs (Stanford, Berkeley, etc.) won't have as many pure probability faculty. Mathematics departments tend to have more probability faculty. So that is also something for you to consider if you are very sure that you want to be a probabilitist. 

Unfortunately, I am not able to gauge your chances for math departments -- you could check with people on the mathematicsgre forum for some suggestions. 
I think your chances are above average for some of the Statistics departments I mentioned (UNC-Chapel Hill, Michigan State University, University of Minnesota, University of Wisconsin), and you could find a PhD supervisor who is more on the probability side of things. The very top Statistics programs also have a lot of probability theory faculty (Stanford, University of Chicago, and UC Berkeley come to mind), but unless you're ranked in the top 10 or so students in your class, these schools might be hard for you to get into.

I just had an interview with a professor at MSU and I was told to stay tuned since I have a high chance.  Expecting to receive a fast update upon the committee's decision.

truly an asset to this community

Just wanted to give a quick shoutout to @bayessays for being around and giving solid input/advice to a lot of people on here.

Congratulations to both of You. My friend has also finally got the offer. Just yesterday she was sent an letter from director of graduate studies mentioning about her employment and stipend amount and many other related things. 

Congratulations to both of You. My friend has also finally got the offer. Just yesterday she was sent an letter from director of graduate studies mentioning about her employment and stipend amount and many other related things. 

Congratulations! I got the offer too. Btw, do we still have to wait for the official admission letter since the email only said "The Graduate Committee of the Department of Statistics and Actuarial Science has recommended that you be admitted"? Let me know if you got the same email.

Being interviewed is always good.  I wouldn't overthink things.
 
 

Thanks, @StatB, and @shyburrito! By the way, anyone still waiting without any news from UNC Biostat phd? I guess they are delaying decisions these days due to funding issues.

Okay. I hope that you get admitted soon. Regarding funding I guess you will get full funding if you are accepted. 

Oh. This seemed really odd to me. Anyway, let us know how it goes after the interview.
My friend ultimately replied to them very positively and asked about funding. She was told that she will be given full funding and they will send her the offer letter sometime in next week.

That is how I would interpret it, but nothing is definite.  A lot of smaller departments do some vetting to see if applicants are really interested, but they would only do that for someone they're interested in.  I had a weird experience once where the offer didn't pan out after this, but I would suspect that your friend could go to Iowa if they want.

Yes, of course the additional coursework will help, but I don't think you are going to improve your profile by a *drastic* amount.  For instance, if you didn't get into any top 30 programs, I certainly think it's unlikely that you'll be getting into a top 10 program in the next application cycle.  You may be able to move from the 40s to the 30s or 20s, but I wouldn't expect magical results from taking a few more classes.  We have somewhat similar profiles (good undergrad school but mediocre GPA, data scientist at FAANG, and I did submit a nearly perfect GRE score) -- and I didn't get into schools like Ohio State and Purdue after completing an MS with a 3.9 gpa from a top 20 program with all of the PhD coursework and multiple first-author publications.  It may be that the department you were admitted to is an appropriate rank for someone with your background.

Saw this virtual panel apparently from UC-Berkeley and Michigan Stat PhD students to answer questions about Stat PhD admissions and whatnot, and figured people who come here might also be interested in it.  I claim no knowledge of these folks, but Rob Santos tweeted this out, so I'll take the liberty of assuming he hasn't gotten hacked or whatnot (and I haven't seen anyone else post it; if they have, we can delete this).
Anyways, the panel is at 7 PM Eastern time on Oct 13 (today/tomorrow, depending on how you delineate that).  Here's the link:  https://www.statsphd.com/ to their site, or the Zoom registration directly to save you a click:  https://umich.zoom.us/webinar/register/WN_R9Bjv3IdRoqmrdygdnkNPA 

Oh, I didn't realize you were from ISI. ISI enjoys a very strong global reputation, and adcoms will be familiar with the grading situation there. So pedigree is not an issue in your case.
That said, most Statistics PhD students from ISI have a Masters degree in Statistics from ISI, so it's hard to give advice to a math major from there -- though there are a lot of international students in Statistics PhD programs in the U.S. who studied pure math too, so I don't think that would be a handicap. It's just that I'm not sure how your profile would be compared to other ISI students who have actual BSc and MS degrees in Stat.
I think (?) you might have a chance at schools like schools in the range of University of Florida through Michigan State. MSU in particular has some great probability theory people. But it might make more sense for you to talk with an academic counselor at your institution to see how you stack up against your ISI peers who have BSc and MS in Statistics.

Hi @Riemannsum. If I am not wrong you have done Bachelor of Mathematics (BMath) and presently doing MMath(Master of Math) at the Bangalore center of ISI. @Stat Assistant Professor has already given you some very good advice. You should definitely approach some professors in stat-math unit who are close to you.  I have seen some past students from MMath (whose interest was mainly skewed towards probability) getting accepted at UNC, MSU for their PhD in Statistics. So, perhaps you can check that and can try to contact those seniors.(If you need more help you can message me). 

Most Stat PhD students from India are from Indian Statistical Institute or University of Calcutta, and occasionally some are from IIT (though in that case, they usually do a Masters in Statistics somewhere first). Since your profile is not "typical" for a Statistics PhD student in the U.S. who is from India, it might be hard to gauge your chances.
Does your program have a track record of success in placing its students in PhD programs in the U.S.? Have graduates from your program successfully been admitted to Statistics PhD programs in the U.S. in the past? The GPA may also need some context. At ISI, it is known that anything over 80 is considered very good. Is a 74-ish considered "above average" at your institution? 

Thanks for the clarification. I think you should aim for schools around the general tier of Ohio State, UIUC, University of Florida, Rutgers. I think you would have a decent shot at those schools. You can apply to a few that are ranked above that (as you correctly conjectured in your original post, your chances would be better at large state school programs) and a few ranked below 40 like UConn, UIowa, etc. and maybe a few even below that just for good measure.

What does your percentage at ISI roughly translate to on a 0.0-4.0 GPA scale? From what I gather, it isn't like 90-100 is an A, 80-90 is a B, etc. like in the U.S., because I saw that somebody who had 88 percent from ISI graduated "First Class with distinction," and they were admitted to UPenn Wharton. So is anything over 80 considered an A at ISI? I wouldn't be surprised if an 80 is considered very good at ISI given how rigorous this institution is. 
 With a bit more context about your academic performance (especially in comparison to your peers at ISI), we can give more tailored advice. Thanks!