(b) A Roulette table has numbers from 0 to 36 to place bets on. 18 of them red, 18 of them are black, and one of them is green. In every round, one of the numbers between 0 and 36 is randomly chosen to win. You can place a bet on a single number, or you can place a bet on all red numbers simultaneously. If you place $1 on red and a red number is chosen, you double your money, otherwise you lose your money. If you place $1 on a single number and this number is chosen, you get back $36 (including the $1 you placed), otherwise you lose your money. (i) Calculate the expected value for the money you win if you place $1 on red. What is the expected value for the money you win if you place $1 on a single number? Show all your working. (ii) How much money can you expect to end up with, if you start out with $37 and you place $1 on red 37 times in a row? Justify your answer. (iii) Assume you start with $10 and you place $1 on a single number 10 times in a row. Calculate the probability that you lose all of your money. What is the probability that you win exactly once? Show all your working and justify your answers.

A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Expected value
(b) A Roulette table has numbers from 0 to 36 to place bets on. 18 of them red, 18 of them are black,
and one of them is green. In every round, one of the numbers between 0 and 36 is randomly chosen to
win. You can place a bet on a single number, or you can place a bet on all red numbers simultaneously. If
you place $1 on red and a red number is chosen, you double your money, otherwise you lose your
money. If you place $1 on a single number and this number is chosen, you get back $36 (including the $1
you placed), otherwise you lose your money.
(i) Calculate the expected value for the money you win if you place $1 on red. What is the expected
value for the money you win if you place $1 on a single number? Show all your working.
(ii) How much money can you expect to end up with, if you start out with $37 and you place $1 on red
37 times in a row? Justify your answer.
(iii) Assume you start with $10 and you place $1 on a single number 10 times in a row. Calculate the
probability that you lose all of your money. What is the probability that you win exactly once? Show
all your working and justify your answers.
Transcribed Image Text:(b) A Roulette table has numbers from 0 to 36 to place bets on. 18 of them red, 18 of them are black, and one of them is green. In every round, one of the numbers between 0 and 36 is randomly chosen to win. You can place a bet on a single number, or you can place a bet on all red numbers simultaneously. If you place $1 on red and a red number is chosen, you double your money, otherwise you lose your money. If you place $1 on a single number and this number is chosen, you get back $36 (including the $1 you placed), otherwise you lose your money. (i) Calculate the expected value for the money you win if you place $1 on red. What is the expected value for the money you win if you place $1 on a single number? Show all your working. (ii) How much money can you expect to end up with, if you start out with $37 and you place $1 on red 37 times in a row? Justify your answer. (iii) Assume you start with $10 and you place $1 on a single number 10 times in a row. Calculate the probability that you lose all of your money. What is the probability that you win exactly once? Show all your working and justify your answers.
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