a marksman scores at each shot either 10, 9, 8, 7 or 6 with respective probabilities 0.5, 0.3, 0.1, 0.05, 0.05.

Question

He fires 100 shots. What is the probability that: a. his aggregate score exceeds 980? b. his aggregate score exceeds 950? Thanks~

Answer 1

His aggregate score can be written as $X:=\sum_{i=1}^{100}X_i$ where $X_i$ denotes his score that the $i$-th shot.

Here the $X_i$ are iid.

You can find mean by applying linearity of expectation and symmetry:$$\mathbb EX=100\mathbb EX_1$$

Because there is independence you can find $\mathsf{Var}(X)$ as summation of the variances, leading to:$$\mathsf{Var}(X)=100\mathsf{Var}(X_1)$$

Then for standard deviation we find:$$\sigma_X=\sqrt{\mathsf{Var}(X)}=10\sigma_{X_1}$$