Graphing not required

We can consider $x\in[0,360)$ , as after that the values of $\sin(x^{\circ})$ begin repeating themselves.

For $x\in[30,150]$ , which consists of $150-30+1=121$ numbers, $\sin(x^{\circ})\ge\frac{1}{2}$ .

Hence the number of cases for which $\sin(x)<\frac{1}{2}$ will be $360-121=239$ .

Hence the probability, $\frac{239}{360}$ .

Note: I'm not so sure about taking $x\in[0,360)$ . The answer would not change if we took $x\in(0,360]$ , but should one of those sets be considered or should we consider $x\in[0,360]$ ? My argument is that if we take $x\in[0,360]$ , the next set in order would be $x\in[361,542]$ , and as you can see, we can arrive to a contradiction. Please comment your arguments or agreements, if any.