This means the expected number of solves for you to get a success is 2.25 solves. The standard deviation is sqrt[(5/9) / (4/9)^2] = 3sqrt(5)/4 = 1.68 solves.
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As for your next success I predict you will get one in the next 2.25 +- 1.68 solves
Typically in statistics you don't want to use an interval of one standard deviation, because that will only encompass about 68% of the possibilities. A 90% confidence interval (that is, the random variable you are studying has a 90% chance of being in that interval) would be more relevant. Since the SD is so high and Mike can't get a success before his first attempt, the appropriate 90% confidence interval value would be an interval from 1 to 2.25 + 1.28 * SD, or from 1 to 4.4. So we can expect, with 90% confidence, that Mike will get a success at least in his next 4.4 tries.