The key thing to notice here is that we are dealing with probability with replacement. Each time we take a ball out of the bag, we put it back in the bag before taking out another ball. This means that the two events are independent - what happens on the first draw doesn't affect the probability of the second draw.
Our first step is to combine our ratios - we know how bag P compares to Q and how Q compares to R, but we don't know how P compares to R. The way we do this is by making Q the same number in each ratio.
Ratios work just like fractions in that we can multiply both sides of the ratio by the same number and still have the same ratio - 1:2 is the same as 2:4 is the same as 3:6 is the same as 150:300.
Our ratios are
P:Q
2:3
Q:R
4:5
The Least Common Multiple of 3 and 4 is 12, so let's make Q = 12 in both ratios.
P:Q
2:3 --- 2*4:3*4 --- 8:12
Q:R
4:5 --- 4:*3:5:*3 --- 12:15
Since Q is now the same in the two ratios, we can combine them.
P:Q:R
8:12:15
Since we don't really care which bags the balls came from, let's rewrite this in terms of color.
red:blue:white
8:12:15
1. The probability of picking a red ball out of all of the balls (all red, blue, and white balls) is 8 / (8+12+15) = 8 / 35.
2a. There are two possibilities for how we can pick a white ball and a blue ball: 1) we can pick a white ball first and then a blue ball, and 2) we can pick a blue ball first and then a white ball. The probability of 1) is 15/35 * 12/35 = 3/7 * 12/35 = 36/245, and the probability of 2) is 12/35 * 15/35 (the reverse) = 36/245. Adding the two probabilities together gives 36/245 + 36/245 = 72/245.
2b. Specifying the order in which the balls are picked limits our possibilities. There is a 12/35 chance we will pick a blue ball first and an 8/35 chance we will pick a red ball second - 12/35 * 8/35 = 96/1225.
