Mathematics Write-up
Spinner and Card Game
Problem Statement:
In this problem we were asked to do two things. The first problem said that two people are playing a game with a spinner that is divided so that ⅕ of the time, it will land on Al’s area, and ⅘ of the time, it will land on Betty’s. When the spinner lands on Al, he gets $1.25 and when the spinner lands on Betty, she gets $0.30. So, what is Al and Betty’s expected value per spin? How can the payments be changed so the game is more fair?
The second game was involving cards. Two people are playing a game in which every time a jack is picked, player B pays Player A $0.20 and every time a heart is drawn, Player A pays Player b $0.08. If neither a jack or heart is picked, both players give a penny to charity. What is the expected value for all three players (Player A and B, and the charity)?
Process:Describe what you did in attempting to solve the problem. Do this part even if you did not solve the problem.
Well, in order to find theoretical probability, I divided up the spinner and count them. Then we decided which sections were Betty’s and which ones were Al’s. For theoretical probability my partner and I spun the spinner 25 times and we recorded who won how many times based on if they played 25 times. For observed probability, all we had to do was put each player’s winnings over 25 to make a fraction. This would be their observed chance of winning the game. Finally, for expected value, we took the probability of both players and multiplied it by how much money they would win if they did land on their own spinner.
Solution:State the solution clearly as you can. Explain how you know your answer is correct.
Theoretical Probability
P(Al’s section) = 1/5
P(Betty’s section) = 4/5
Observed Probability:
P(Al’s section) = 4/25
P(Betty’s section) = 21/25
Theoretical Weighted Probability:
P(Al) = 1/5(1.25) = $0.25
P(Betty) = 4/5(.30) = $0.24
Observed Weighted Probability:
P(Al) = 4/25(1.25) = $0.2
P(Betty) = 21/25(.30) = $0.252
Connection:
What is the relationship between Expected Value, Observed Weighted Probability, & Theoretical Weighted Probability? What are some real-life examples in which these concepts are being used?
Well, weighted probability can be related to both theoretical and observed probability. Theoretical probability is the chance of something happening that is calculated in a mathematical way. Observed probability is the factual answer, something that cannot be controlled by the viewer. Finally, Weighted probability is the chance of something with a lower probability ending up with a greater or more positive outcome in the long run! These probabilities are often used in real life, the most common would be in casinos and gambling.
Evaluation:Discuss your personal reaction to the problem.
I liked this activity because it was not only challenging, but it taught us about all different kinds of probabilities in a way that is much easier to understand than just a plain old lecture. One habit of a mathematician that I used was definitely used was staying organized. There was a lot of recording something and going back to refer to the same information so it was hard to keep everything organized. I wouldn’t change this problem, I think that this was the perfect level for our class, it was more challenging than easy but once you got a hang of the concept it got much easier to do. I really did enjoy this project and I hope that we have more activities involving games in the future.
Spinner and Card Game
Problem Statement:
In this problem we were asked to do two things. The first problem said that two people are playing a game with a spinner that is divided so that ⅕ of the time, it will land on Al’s area, and ⅘ of the time, it will land on Betty’s. When the spinner lands on Al, he gets $1.25 and when the spinner lands on Betty, she gets $0.30. So, what is Al and Betty’s expected value per spin? How can the payments be changed so the game is more fair?
The second game was involving cards. Two people are playing a game in which every time a jack is picked, player B pays Player A $0.20 and every time a heart is drawn, Player A pays Player b $0.08. If neither a jack or heart is picked, both players give a penny to charity. What is the expected value for all three players (Player A and B, and the charity)?
Process:Describe what you did in attempting to solve the problem. Do this part even if you did not solve the problem.
Well, in order to find theoretical probability, I divided up the spinner and count them. Then we decided which sections were Betty’s and which ones were Al’s. For theoretical probability my partner and I spun the spinner 25 times and we recorded who won how many times based on if they played 25 times. For observed probability, all we had to do was put each player’s winnings over 25 to make a fraction. This would be their observed chance of winning the game. Finally, for expected value, we took the probability of both players and multiplied it by how much money they would win if they did land on their own spinner.
Solution:State the solution clearly as you can. Explain how you know your answer is correct.
Theoretical Probability
P(Al’s section) = 1/5
P(Betty’s section) = 4/5
Observed Probability:
P(Al’s section) = 4/25
P(Betty’s section) = 21/25
Theoretical Weighted Probability:
P(Al) = 1/5(1.25) = $0.25
P(Betty) = 4/5(.30) = $0.24
Observed Weighted Probability:
P(Al) = 4/25(1.25) = $0.2
P(Betty) = 21/25(.30) = $0.252
Connection:
What is the relationship between Expected Value, Observed Weighted Probability, & Theoretical Weighted Probability? What are some real-life examples in which these concepts are being used?
Well, weighted probability can be related to both theoretical and observed probability. Theoretical probability is the chance of something happening that is calculated in a mathematical way. Observed probability is the factual answer, something that cannot be controlled by the viewer. Finally, Weighted probability is the chance of something with a lower probability ending up with a greater or more positive outcome in the long run! These probabilities are often used in real life, the most common would be in casinos and gambling.
Evaluation:Discuss your personal reaction to the problem.
- What did you learn from it?
- Describe one Habit of a Mathematician that you used?
- How would you change the problem to make it better?
- Did you enjoy working on it?
- Was it too hard or too easy?
I liked this activity because it was not only challenging, but it taught us about all different kinds of probabilities in a way that is much easier to understand than just a plain old lecture. One habit of a mathematician that I used was definitely used was staying organized. There was a lot of recording something and going back to refer to the same information so it was hard to keep everything organized. I wouldn’t change this problem, I think that this was the perfect level for our class, it was more challenging than easy but once you got a hang of the concept it got much easier to do. I really did enjoy this project and I hope that we have more activities involving games in the future.