## 8.9 Box

### 8.9.1 Box Problem #01

In a box, there are 5 blue marbles, 7 green marbles, and 8 yellow marbles. If two marbles are randomly selected, what is the probability that both will be green or yellow, if taken

1. with replacement

2. without replacement

• Correct or not: $$\frac{^7C_2}{^{20}C_2}+\frac{^8C_2}{^{20}C_2}$$
• $$\frac{^7C_1 \times ^7C_1}{^{20}C_1 \times ^{20}C_1}+\frac{^8C_1 \times ^8C_1}{^{20}C_1 \times ^{20}C_1}$$
• Without replacement: $$\frac{^7C_1 \times ^6C_1}{^{20}C_1 \times ^{20}C_1}+\frac{^8C_1 \times ^7C_1}{^{20}C_1 \times ^{20}C_1}$$

### 8.9.2 Box Problem #02

There are some balls in a box as below

Color # Balls
White 3
Black 6
Red 7
Green 5
Yellow 4
Violet 9
Blue 8

If three balls are drawn at random, what is the probability there are all red or green?

• $$\frac{^7C_3}{^{42}C_3}+\frac{^5C_3}{^{42}C_3}$$
• 0.039

### 8.9.3 Box Problem #02

There are 9 red and 7 white balls in a box. 6 balls are picked randomly. What is the probability that 3 balls are red and 3 balls are white?

• $$\frac{^9C_3 \times ^7C_3}{^{16}C_6}$$
• $$\frac{^9C_3}{^{16}C_3} \times \frac{^7C_3}{^{16}C_3}$$
• $$\frac{^9C_3}{^{16}C_3} + \frac{^7C_3}{^{16}C_3}$$
• $$\frac{^9C_3}{^{16}C_6} \times \frac{^7C_3}{^{16}C_6}$$
• Whatever we draw together will be in $$r$$ in $$^nC_r$$
• Answer: $$\frac{^9C_3 \times ^7C_3}{^{16}C_6}$$=0.367