You play the following simple game of chance. A fair coin is flipped. If it comes up heads, you win a dollar. If it comes up tails, you lose a dollar. Suppose you start with N dollars in your pocket. You play repeatedly until you either reach M dollars or lose all your money, whichever comes first. M and N are fixed positive integers such that 0 < N < M. a. Show that with probability one the game ends, in other words, that the amount of money in your pocket will eventually hit 0 or M.
b. What is the probability that the game ends with M dollars in your pocket?

A) E(x) = n ( 1 - p ) = u ( 1 - 1 ) =0

B)

Step-by-step explanation:

In other to make money ( M ) you have to flip more head than tail

where : 0 < N < M

N = dollars in your pocket

M = money made from flipping the coin

A) Show that with the probability one that the money in your pocket ends in a 0 or M  as the game ends

This is a binomial distribution problem hence to show that the money in your pocket ends in a 0 or M can be shown as

E(x) = np = u * 1

p = p( head )

1 - p = P ( tail ( failure ))

Hence

E(x) = n ( 1 - p ) = u ( 1 - 1 ) =0

b) probability that the game ends with M dollars in your pocket

To end up with M dollars you have to flip a head more often than a tail

P( head ) = p

P( tail ) = 1 - p

Hence the probability that the game ends with M dollars in your pocket

=

n = number of successions

p = probability of flipping a head

p-1 = probability of flipping a tail

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12

coins of silver or gold

step-by-step explanation: