A simple coin game is as follows: we have a box, which starts empty. P1 flips a fair coin. If it comes up heads, P2 gets the contents of the box, and the game ends. If it comes up tails, P1 puts a dollar in the box and they flip again; this repeats until it comes up heads. a. With what probability will P2 win exactly 10 units?
b. Write ğ[infinity] = Ã[infinity] ğ=0 ğ ğ . Show that (1 â ğ)ğ[infinity] = 1, so that ğ[infinity] = 1 1 â ğ .
c. Show that âï¸[infinity] ğ=0 ğğğ = âï¸[infinity] ğ=0 ğ ğ ! + ğ âï¸[infinity] ğ=0 ğ ğ ! + ğ 2 âï¸[infinity] ğ=0 ğ ğ ! + · · · (look carefully at the limits of the sums!) and so show that âï¸[infinity] ğ=0 ğğğ = ğ (1 â ğ) 2 .
d. What is the expected value of the game? (you may find the results of the two previous subexercises helpful; theyâre not there just for show).
e. How much should P2 pay to play, to make the game fair?

529 and 531

step-by-step explanation:

{x = -1/2 ,   y = -1

step-by-step explanation:

solve the following system:

{2 x + y = -2

{10 x - 3 y = -2

in the first equation, look to solve for y:

{2 x + y = -2

{10 x - 3 y = -2

subtract 2 x from both sides:

{y = -2 x - 2

{10 x - 3 y = -2

substitute y = -2 x - 2 into the second equation:

{y = -2 x - 2

{10 x - 3 (-2 x - 2) = -2

10 x - 3 (-2 x - 2) = (6 x + 6) + 10 x = 16 x + 6:

{y = -2 x - 2

{16 x + 6 = -2

in the second equation, look to solve for x:

{y = -2 x - 2

{16 x + 6 = -2

subtract 6 from both sides:

{y = -2 x - 2

{16 x = -8

divide both sides by 16:

{y = -2 x - 2

{x = -1/2

substitute x = -1/2 into the first equation:

{y = -1

{x = -1/2

collect results in alphabetical order:

answer:   {x = -1/2 ,   y = -1

solve systems of equations by graphing. a system of linear equations contains two or more equations e.g. y=0.5x+2 and y=x-2. the solution of such a system is the ordered pair that is a solution to both equations. to solve a system of linear equations graphically we graph both equations in the same coordinate system