I was watching a Masahiro Tanaka game last year and he got out of a men on 2nd and 3rd with no outs jam by striking out the first two batters. He just kept throwing his "out pitch" and they just kept missing exactly the same way. And all the batters knew that even just the runner on 3rd scoring is valuable cause it was 1-1 or 1-0 or 0-0 (I forgot) after a few innings. What's the chance of an average MLB SP doing that?
4 thousandths of a percent...ish.
6 swings and misses on 6 pitches in a row against the same pitch. For that we should talk about whiff%.
As a commenter points out, it's very difficult to answer this in the general case as each pitcher is very different in how they get outs (some are strikeout pitchers, other are contact pitchers, some pitchers generate swinging strikes, others more called strikes). However, for the sake of the exercise, let's look at Tanaka. If we wanted to do, we could do another exercise and use league average swStr% from fangraphs and determine the likelihood for a league average hitter against a league average pitch, but that's less interesting I think and certainly much less likely than using someone like Tanaka who has a plus swing and miss pitch.
First, because it would be a very small sample, I'm going to ignore both count and game situation and assume all pitches are thrown in a vacuum. This is silly, but the numbers would quite quickly break down to small enough increments to be very unreliable. I'm also going to use the 2014 numbers as it's a much larger sample than this year (and happens to be the year that you're looking for).
First of all, we need to figure out which pitch Tanaka was throwing, based the swing and miss numbers, I'm going to guess it was probably his Four Seam fastball. This pitch generated the most swings and misses for Tanaka last year.
Fangraphs, conveniently provides swinging strike numbers. As we've set up our experiment to assume that swinging strikes are independent of each other (that's probably unlikely, but our numbers don't control or that, and if anything I'd believe this ends up coming out low).
Tanaka's Four Seamer was very good last year. 27.4% of the time he threw it last year, he got a whiff.
This means that the expected numbers of swinging strikes in 6 Four Seam fastballs would only be 1.64.
However, that's not what happened. Tanaka was particularly nasty and got 6 swinging strikes in a row.
The probability of an independent event happening 6 times in a row is the probability of the event to the power of the instances. Thus, in this case:
.274^6 = .0000423
So the odds of Tanaka throwing, and getting 6 strikes on the same pitch in a row is somewhere around 4 thousandths of a percent.
The overall MLB batting average was .251 last season. You can see that this is fairly typical average if you check the other years. Actually, 2014 seems to be a bit lower than other years, some seasons had a league average higher than .260.
Anyways, lets just round to .250 for easy math. That means each batter typically has a 25% chance of getting a hit or 75% chance of getting out.
According to the same website, that same season teams averaged 1248 strikeouts. There are 162 games in a season, so that works out to 7.7 strikeouts per game. Granted there are sometimes games where teams only get 8 innings to bat, or games that go into extra innings, but for simplicity, lets just assume 9 innings per game, which means 27 outs per game. That is 7.7/27 = 28.5% of the outs in a game will be strikeouts.
So if you assume that 75% of batters will get out, and that 28.5% of those outs will be strikeouts, you basically have a 21.375% chance of striking out any individual batter.
The odds of striking out two batters in a row is therefore 4.6%.
To figure out how likely it would be to strike out two batters with them swinging at 6 pitches in a row (3 each), you'd have to figure out what the odds are that they would swing (and miss) at each pitch. I have no idea where you might find the numbers for that, but I imagine it would very low odds.