We can use the binomal probability distribution, and ignore the rules of baseball:
1. Assuming that each at bat results in either a hit or an out, then the probability of a hit is p=0.34, and the probability of an out is q=1-0.34 = 0.66. (Note - In baseball an at bat often results in an outcome that isn't counted in the batting average - such as getting a walk, hit by pitch, reaching base on an error, and sacrifice fly. But we'll ignore that for those possibilities for the purposes of solving this problem.)
2. Over 200 at bats the mean for the expected number of hits is 0.34 x 200 = 68.
3. The standard deviation of a binomial distribution is
4. If he gets only 60 hits, that would be 8 away from the mean, or 8/(6.7) = 1.19 standard deviations below the mean.
You can use the cumulative distribution tables to find the probability of him achieving less than -1.19 sigma.