There are three contestants in a game. The game is played as follows: 3 tokens that are mumbered 1, 2, and 3 are placed in a box. Each contestant selects a token with replacement (i.e, contestants select a token, announces the mumber and places the token back to the box). The person(s) who picked the largest number share a cash prize of $300 equally among them. (a) Define the sample space representing the game. (b) What is the probability that winners receive $100 cach. (c) What is the probability that there is a single winner who takes the $300? (d) How do your answers to parts (b) and (c) change when tokens are selected without replacement?
There are three contestants in a game. The game is played as follows: 3 tokens that are mumbered 1, 2, and 3 are placed in a box. Each contestant selects a token with replacement (i.e, contestants select a token, announces the mumber and places the token back to the box). The person(s) who picked the largest number share a cash prize of $300 equally among them. (a) Define the sample space representing the game. (b) What is the probability that winners receive $100 cach. (c) What is the probability that there is a single winner who takes the $300? (d) How do your answers to parts (b) and (c) change when tokens are selected without replacement?
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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Please answer correctly and show all your work. Attached is the formula sheet you can use.
Transcribed Image Text:There are three contestants in a game. The game is played as follows: 3 tokens that are numbered 1,
2, and 3 are placed in a box. Each contestant selects a token with replacement (i.e, contestants select
a token, announces the mumber and places the token back to the box). The person(s) who picked the
largest number share a cash prize of $300 equally among them.
(a) Define the sample space representing the game.
(b) What is the probability that winners receive $100 each.
(c) What is the probability that there is a single winner who takes the $3002
(d) How do your answers to parts (b) and (c) change when tokens are selected without replacement?
Transcribed Image Text:Axioms of Probability
Also Note:
1. P(S)=1
For any two events A and B,
2. For any event E, 0< P(E) < 1
P(A) = P(AN B) + P(AN B')
3. For any two mutually exclusive events,
and
P(EUF) = P(E)+ P(F)
P(AN B)
P(A|B)P(B).
Events A and B are independent if:
Addition Rule
P(EUF) = P(E)+ P(F) – P(ENF)
Р(AВ) — Р(А)
or
Conditional Probability
Р(An B) %3D P(A)Р(В).
Р(BJA) —
P(ANB)
P(A)
Bayes' Theorem:
Total Probability Rule
Р(АВ)P(B)
Р(А|B)Р(В) + Р(A|B')P(В')
P(A)-P(A | Β) P(Β) + P(A| Β') P(Β')
P(B|A)
Similarly,
Similarly,
Р(A) —DР(A|E,)Р(E) + P(A|E2) Р(E)+
P(B|E1)P(E1)
P(B|E1)P(E1)+P(B|E2)P(E2) + · · · + P(B|Er)P(Ex)
...+ P(A|Ek)P(Ek)
P(E1\B)
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