[/quote]
,
and 
are given on side 
of convex quadrilateral 
(with 
)
and 
.
.
where 
.
such that 
be nonnegative real numbers such that 
.
and 
Prove that
Where 
Y by
Alice and Bob play a game where Alice starts with 
MathJuice bottles and Bob starts with 
MathJuice bottles. An unfair coin is then flipped, with probability 
of landing heads. If the coin lands heads, Alice gives Bob a bottle; otherwise, Bob gives Alice a bottle. This process repeats until someone runs out of bottles.
(a): What is the probability that Bob will lose all of his bottles before Alice does?
(b): What is the expected number of times the coin has been flipped by the time the game ends?
Source: Own
MathJuice bottles and Bob starts with 
MathJuice bottles. An unfair coin is then flipped, with probability 
of landing heads. If the coin lands heads, Alice gives Bob a bottle; otherwise, Bob gives Alice a bottle. This process repeats until someone runs out of bottles.(a): What is the probability that Bob will lose all of his bottles before Alice does?
(b): What is the expected number of times the coin has been flipped by the time the game ends?
Source: Own
This post has been edited 1 time. Last edited by Bummer12345, Mar 20, 2025, 1:49 AM
be the probability that Bob loses all of his bottles before Alice does, if Alice starts with 
bottles and Bob starts with 
bottles. We have 
,
into the formula for 
,
,
and conclude that 
.
,
.
flips for the game to end.
