Binomial distribution. Exactly 7 successes in 8 tries. Say the probability of success was s, where s is 1/(square root of 10), a figure a little bit above .300. Probability = 8*s^7(1-s), which is somewhere around .15%. Changing the question to "... at least 7 successes" wouldn't affect the outcome much.
However, that's not the right sample to analyze. My only personal memory of Yastrzemski in a clutch situation is a weak pop-up to end a one-game playoff in 1978. At a minimum, you'd need to look at his post-season games as well as his heat-of-the-pennant-race games, and look at his overall batting average in those. I think you'd probably find it to be a LOT below .875, and indeed well below .400 ...
http://www.baseball-almanac.com/players/playerpost.php?p=yastrca01&ps=ws has his World Series percentage: .352.
Overall postseason: .369. Even better.
http://www.baseball-reference.com/y/yastrca01.shtml
And raise that very slightly for the Bucky Dent game, his game-ending popout notwithstanding:
http://www.baseball-almanac.com/box-scores/boxscore.php?boxid=197810020BOS
Better than his regular-season batting, actually. But not "unstoppably clutch" either, of course. No batter is.