. Suppose an urn contains 20 colored balls: 10 white, 7 red, and 3 blue. You are going to draw some balls out of the urn, and you will win if you obtain certain combinations of the balls.

a. Find the probability that you draw a red, white, and blue ball out of the urn in that exact order if you are taking three draws.

b. Find the probability that you draw a red, white, and blue ball out of the urn inanyorder if you are taking three draws.

c. Suppose that the red balls are the "Danger" balls, and, if you draw any red ball out of the urn, the game is over.

Find the probability that you will draw 5 balls out of the urn without drawing a "Danger" ball. Based on your computation, would you be willing to bet on this particular game (assuming the game pays at 1-to-1 odds)? Why or why not?

2. The Illinois Lottery sponsors a daily game called "Pick 4", where players pick a 4-digit number (digits from 0-9) and try to match the winning number drawn.

There are two common bets that one can make:

- Straight: The player must match the numberexactlyas drawn (e.g., if the winning number is 8315, but the player has 8351, the playerloses).

- Box: The player only needs to match the digits of the drawn number in any order (e.g., if the winning number is 8315, but the player has 8351, the playerwins).

a. What is the probability a player wins the "Straight" bet?

b. What is the probability a player wins the "Box" bet under the following conditions:

i. All four digits the player picks are different

ii. The player has three different digits, with the fourth being a repeat of one of the other three

c. If "Pick 4" was a fair game, how much should a player win in the followingcases:

i. Player bets $1 on a "Straight" ticket

ii. Player bets $1 on a "Box" ticket, with no repeated digits

NOTE: The Illinois Lottery pays $5,000 for a "Straight" ticket win, and $200 for a "Box" win if all four digits are different (minus the $1 originally bet in each case; this isdifferentfrom casino games!).

d. Suppose you could win the $5,000 "Straight" prize by matching either 3 digits or 4 digits exactly. Does this change the probability of winning into a game that's worth playing? Why or why not?