In Monopoly, your token is allowed to leave the "jail" cell if you roll doubles: you roll two 6-sided dice and each shows the same face. Zach hates being in jail, because it reminds him of watching "The Wire" on TV. So he invents a couple of weighted dice that are not independent. In particular, if you roll either die on its own it's a fair die: each outcome has probability 1/6. But if you roll one die and then the other, the red die will take the same outcome as the blue die exactly half the time: all other outcomes are equally likely. (a) Suppose you roll a 3 on the blue die. What is the probability distribution of the red die given this outcome on the blue die? (b) Find the full probability distribution for the value of the sum of the two faces of the dice. (c) What is the probability you roll doubles? (d) What is the probability that you roll a 7 as the sum of the two dice?

A blood bank needs 9 people to help with a blood drive. 12 people have volunteered. Find how many different groups of 9 can be formed from the 12 volunteers.

Step-by-step explanation:

A blood bank needs 9 people to help with a blood drive. 12 people have volunteered. Find how many different groups of 9 can be formed from the 12 volunteers.

geometric series converge the common ratio r has magnitude less than one.

which is equivalent to

for the example we need

|2-x| < 1

-1 < 2-x   and 2-x < 1

x < 3 and x > 1

we can write that

answer: 1 < x < 3

1a. common ratio r=250/25=10 > 1     diverges

1b.   r=50/100=1/2 converges

when these converge they converge to

where a is the first term.

answer: 200

1c. r=-1   so it's not the case that |r| < 1.     diverges

2a. x is the common ratio so we converge when |x|< 1

2b.   r=-10x / -2 = 5x.

we converge when

|5x| < 1

|x| < 1/5

2c.   r=3-2x

-1 < 3-2x and 3-2x < 1

-4 < -2x and 2 < 2x

x < 2 and 1 < x

1 < x < 2

2d. [/tex] r=-48x^3/12 = 4x^3[/tex]

|4x^3| < 1

|x^3| < \frac 1 4

|x| < 4^{-1/3}