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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?

Respuesta :

batolisis

Answer:

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

B) [tex]_{n} C_{x} .P(1-p)_{n-x}[/tex]

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

= [tex]_{n} C_{x} .P(1-p)_{n-x}[/tex]

n = number of successions

p = probability of flipping a head

p-1 = probability of flipping a tail

lhmarianateixeira

In this exercise we have to use the knowledge of probability and binomials to calculate what is asked in this way we find that:

A) [tex]E(x) = n ( 1 - p ) = u ( 1 - 1 ) =0[/tex]

B) [tex]nC_xP(1-p)_{n-x}[/tex]

In other to make money ( M ) you have to flip more head than tail where : [tex]0 < N < M[/tex]

  • 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:

[tex]E(x) = np = u * 1\\p = p( head )\\1 - p = P ( tail ( failure ))\\E(x) = n ( 1 - p ) = u ( 1 - 1 ) =0[/tex]

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

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

[tex]P( head ) = p\\P( tail ) = 1 - p\\nC_xP(1-p)_{n-x}[/tex]

See more about probability at brainly.com/question/795909

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