What is the Probability came from the same machine Machine A produced 65 of the day’s output of Product X and machine B produced the other 55. If three products are selected with replacement at random from the day’s output, the probability that, 
My attempts:
(a) three of the products came from the same machine 
$$\binom{65}{3} + \binom{55}{3} = 69915$$
(b) there was one from machine A and two from machine B
$$\binom{65}{1} + \binom{55}{2} = 1550$$
 A: When you have replace the chosen piece again, each new selection does not depend on your previous choices. So for (a), the probability that all parts came from the same machine is 
$$
\frac{65^3+55^3}{110^3},
$$
and for (b), the probability that one part came from machine A and two from machine B is 
$$
\frac{3 \cdot 65 \cdot 55^2}{110^3}.
$$
Why? For part (a), we can either have all parts from machine A or all parts from machine B. Since these are disjoint events, 
$$
P(\text{all parts from same machine}) = P(\text{all parts from machine A})+P(\text{all parts from machine B}).
$$
There are $65^3$ ways to select all the parts to come from machine A, because we replace the chosen part each time. Similarly, there are $55^3$ ways for choosing all the parts from machine B. Finally, there are 110 parts total, so there is $110^3$ ways to pick 3 with replacement. 
Another way to see that $P(\text{all parts from machine A}) = (65/110)^3$ is that the parts picked first, second and third are all independent from each other (the fact that the parts are picked with replacement is key here). 
