Communities

Writing
Writing
Codidact Meta
Codidact Meta
The Great Outdoors
The Great Outdoors
Photography & Video
Photography & Video
Scientific Speculation
Scientific Speculation
Cooking
Cooking
Electrical Engineering
Electrical Engineering
Judaism
Judaism
Languages & Linguistics
Languages & Linguistics
Software Development
Software Development
Mathematics
Mathematics
Christianity
Christianity
Code Golf
Code Golf
Music
Music
Physics
Physics
Linux Systems
Linux Systems
Power Users
Power Users
Tabletop RPGs
Tabletop RPGs
Community Proposals
Community Proposals
tag:snake search within a tag
answers:0 unanswered questions
user:xxxx search by author id
score:0.5 posts with 0.5+ score
"snake oil" exact phrase
votes:4 posts with 4+ votes
created:<1w created < 1 week ago
post_type:xxxx type of post
Search help
Notifications
Mark all as read See all your notifications »
Q&A

Post History

#3: Post edited by user avatar DNB‭ · 2022-01-17T08:54:59Z (almost 3 years ago)
  • Please see the sentence beside my red line. The notion of a "sanity check" suggests that these resultant integers should be obvious, without calculation or contemplation. But why's it plain and intuitive that $p = 1/2 \implies P(C) = 1/2$?
  • >50. Calvin and Hobbes play a match consisting of a series of games, where Calvin has
  • probability p of winning each game (independently). They play with a "win by two" rule: the first player to win two games more than his opponent wins the match. Find
  • the probability that Calvin wins the match (in terms of p), in two different ways:
  • >
  • >(a) by conditioning, using the law of total probability.
  • >
  • >(b) by interpreting the problem as a gambler's ruin problem.
  • >
  • >## Solution:
  • >
  • >(a) Let C be the event that Calvin wins the match, $X \thicksim Bin(2, p)$ be how many of the first 2 games he wins, and $q = 1 - p$. Then
  • >
  • >![Image alt text](https://math.codidact.com/uploads/k3Z9aeNxmkiVrFNDj8Qx1XNy)
  • Blitzstein, *Introduction to Probability* (2019 2 edn), Chapter 2, Exercise 50, p 94.
  • p 18 in the publicly downloadable PDF of curbed solutions.
  • Please see the sentence beside my red line. The notion of a "sanity check" suggests that these resultant integers should be obvious, without calculation or contemplation. But why's it plain and intuitive that $p = 1/2 \implies P(C) = 1/2$?
  • Indubitably, a game differs from a match. Just because $p = 1/2$ doesn't automatically entail $P(C) = 1/2$.
  • >50. Calvin and Hobbes play a match consisting of a series of games, where Calvin has
  • probability p of winning each game (independently). They play with a "win by two" rule: the first player to win two games more than his opponent wins the match. Find
  • the probability that Calvin wins the match (in terms of p), in two different ways:
  • >
  • >(a) by conditioning, using the law of total probability.
  • >
  • >(b) by interpreting the problem as a gambler's ruin problem.
  • >
  • >## Solution:
  • >
  • >(a) Let C be the event that Calvin wins the match, $X \thicksim Bin(2, p)$ be how many of the first 2 games he wins, and $q = 1 - p$. Then
  • >
  • >![Image alt text](https://math.codidact.com/uploads/k3Z9aeNxmkiVrFNDj8Qx1XNy)
  • Blitzstein, *Introduction to Probability* (2019 2 edn), Chapter 2, Exercise 50, p 94.
  • p 18 in the publicly downloadable PDF of curbed solutions.
#2: Post edited by user avatar DNB‭ · 2021-12-31T08:33:56Z (almost 3 years ago)
  • Please see the sentence beside my red line. The notion of a "sanity check" suggests that these resultant integers should be obvious, without calculation or contemplation. But why's it plain and intuitive that $p = 1/2 \implies P(C) = 1/2$?
  • >50. Calvin and Hobbes play a match consisting of a series of games, where Calvin has
  • probability p of winning each game (independently). They play with a "win by two" rule: the first player to win two games more than his opponent wins the match. Find
  • the probability that Calvin wins the match (in terms of p), in two different ways:
  • >
  • >(a) by conditioning, using the law of total probability.
  • >
  • >(b) by interpreting the problem as a gambler's ruin problem.
  • >
  • >## Solution:
  • >
  • >(a) Let C be the event that Calvin wins the match, $X \thicksim Bin(2, p)$ be how many of the first 2 games he wins, and $q = 1 - p$. Then
  • >
  • >![Image alt text](https://math.codidact.com/uploads/k3Z9aeNxmkiVrFNDj8Qx1XNy)
  • Blitzstein, *Introduction to Probability* (2019 2 edn), Chapter 2, Exercise 50, p 94.
  • p 17 in the publicly downloadable PDF of curbed solutions.
  • Please see the sentence beside my red line. The notion of a "sanity check" suggests that these resultant integers should be obvious, without calculation or contemplation. But why's it plain and intuitive that $p = 1/2 \implies P(C) = 1/2$?
  • >50. Calvin and Hobbes play a match consisting of a series of games, where Calvin has
  • probability p of winning each game (independently). They play with a "win by two" rule: the first player to win two games more than his opponent wins the match. Find
  • the probability that Calvin wins the match (in terms of p), in two different ways:
  • >
  • >(a) by conditioning, using the law of total probability.
  • >
  • >(b) by interpreting the problem as a gambler's ruin problem.
  • >
  • >## Solution:
  • >
  • >(a) Let C be the event that Calvin wins the match, $X \thicksim Bin(2, p)$ be how many of the first 2 games he wins, and $q = 1 - p$. Then
  • >
  • >![Image alt text](https://math.codidact.com/uploads/k3Z9aeNxmkiVrFNDj8Qx1XNy)
  • Blitzstein, *Introduction to Probability* (2019 2 edn), Chapter 2, Exercise 50, p 94.
  • p 18 in the publicly downloadable PDF of curbed solutions.
#1: Initial revision by user avatar DNB‭ · 2021-12-31T08:31:07Z (almost 3 years ago)
How to intuit p = Calvin's probability of winning each game independently = $1/2 \implies$ P(Calvin wins the match) = 1/2?
Please see the sentence beside my red line. The notion of a "sanity check" suggests that these resultant integers should be obvious, without calculation or contemplation. But why's it plain and intuitive that $p = 1/2 \implies P(C) = 1/2$?   

>50. Calvin and Hobbes play a match consisting of a series of games, where Calvin has
probability p of winning each game (independently). They play with a "win by two" rule: the first player to win two games more than his opponent wins the match. Find
the probability that Calvin wins the match (in terms of p), in two different ways:
>
>(a) by conditioning, using the law of total probability.
>
>(b) by interpreting the problem as a gambler's ruin problem.
>
>## Solution:
>
>(a) Let C be the event that Calvin wins the match, $X \thicksim Bin(2, p)$ be how many of the first 2 games he wins, and $q = 1 - p$. Then
>
>![Image alt text](https://math.codidact.com/uploads/k3Z9aeNxmkiVrFNDj8Qx1XNy)

Blitzstein, *Introduction to Probability* (2019 2 edn), Chapter 2, Exercise 50, p 94.    
p 17 in the publicly downloadable PDF of curbed solutions.