The probability of rolling a 4 is ⅙. The probability of another 4 is ⅙²=1/36. A third 4 would be 1/216 and a fourth 4 would be 1/1296. The probability of not rolling a 4 is ⅚, and the probability of no more 4s for the remaining 7 throws is (⅚)⁷. So we have a total probability of 5⁷/6¹¹. But we have to calculate how many ways we can arrange four 4s out of 11: (11)(10)(9)(8)/4!=330.

However, the question asks for no more than four 4s so we must also consider none, 1, 2 and 3 fours.

No fours: (⅚)¹¹

One four: 11(5¹⁰)/6¹¹

Two fours: 55(5⁹)/6¹¹

Three fours: 165(5⁸)/6¹¹

Four fours: 330(5⁷)/6¹¹.

Add these together:

5⁷(625+1375+1375+825+330)/6¹¹=

353906250/362797056=58984375/60466176

(=0.9755=97.55% approx.)