spinner give and take write up!
Spinner Give and Take (Cards Extension)
Al and Betty are playing a game with a spinner. Each time the spinner comes up on the white area (¾ of the circle), Betty wins $1 from Al. When the spinner comes to the shaded region (¼ of the spinner), Al wins $4 from Betty. In the long run and the 100th spin, which player will theoretically have more money?
What we first came up with was that Betty takes ¾ of the circle while Al only takes up ¼. So in one spin theoretically: P(Betty winning)= 75% and P(Al winning)=25%. But we also need to take in consideration that they are both winning different amounts of money. So say they spin the spinner 4 times. Out of those 4 times, the theoretical probability is that the spinner lands on Betty winning $1 three times, and spinner lands on Al once for $4.
So in four spins
Lands On Al
Lands on Betty
9
16
If we spin the spinner 100 times, the theoretical probability is that Al will be in the lead.
¾ of the spins will land on Betty which is 75 spins and ¼ of the spins will land on Al which is 25 spins.
CONNECTION:
The relationship between between Expected Value, Observed Weighted Probability, & Theoretical Weighted Probability is that they all have a chance of becoming a better possibility. Theoretical weighted probability is different than theoretical; probability because the value of the outcomes can go to your favor. Ex. Observed Probability of landing on red is 1/50 and chance of blue is 49/50. So you would want to choose blue. But if red is worth $10,000 and blue is worth $1 you would want to choose the one that will give you the most in the long run which would be red. The weighted probability and expected values all have a factor that affects the values of the outcomes.
Examples of these would be the lottery. The lottery has different values. Lower values have more of a chance of popping up than bigger values. And even though I’m against lottery because everything is sooo fake, that’s how it would work if they actually conducted the lottery fairly.
EVALUATION:
To solve this problem the main habit of a mathematician that I used was breaking it down. At first there was information just directly given and a lot of scenarios you had to solve but taking it part by part and adding it all together in the end really did help. I actually did really enjoy working on it and took a lot from it since I haven’t reviewed probabilities in this way in such a long time. There isn’t anything I would change because the problem was just simple and well brought together. Everything connected right and it wasn’t made up of just busy work. It kinda was pretty easy but that’s just until you actually understand it because the problem does make you think.
EXTENSION:
Archibald and Beatrice play a game in which they draw a card from a deck. Without taking in consideration of who draws the card:
After conducting an actual experiment (25 tests) our results came to be:
Al and Betty are playing a game with a spinner. Each time the spinner comes up on the white area (¾ of the circle), Betty wins $1 from Al. When the spinner comes to the shaded region (¼ of the spinner), Al wins $4 from Betty. In the long run and the 100th spin, which player will theoretically have more money?
What we first came up with was that Betty takes ¾ of the circle while Al only takes up ¼. So in one spin theoretically: P(Betty winning)= 75% and P(Al winning)=25%. But we also need to take in consideration that they are both winning different amounts of money. So say they spin the spinner 4 times. Out of those 4 times, the theoretical probability is that the spinner lands on Betty winning $1 three times, and spinner lands on Al once for $4.
So in four spins
- Betty wins $3 ($1 for the ¾ spins that landed on her region)
- P(Money won out of 4 spins) = $3
- Al wins $4 ($4 for the ¼ spins that landed on her region)
- P(Money won out of 4 spins) = $4
Lands On Al
Lands on Betty
9
16
- Al gets $4 for every spin landed on the shaded region: 9 spins x $4 = $36
- Betty gets $1 for every spin landed on the white region: 16 spins x $1=$16
If we spin the spinner 100 times, the theoretical probability is that Al will be in the lead.
¾ of the spins will land on Betty which is 75 spins and ¼ of the spins will land on Al which is 25 spins.
- Betty wins $75 ($1 for the 75/100 spins she gets)
- Al wins $100 ($4 for the 25/100 spins she gets)
CONNECTION:
The relationship between between Expected Value, Observed Weighted Probability, & Theoretical Weighted Probability is that they all have a chance of becoming a better possibility. Theoretical weighted probability is different than theoretical; probability because the value of the outcomes can go to your favor. Ex. Observed Probability of landing on red is 1/50 and chance of blue is 49/50. So you would want to choose blue. But if red is worth $10,000 and blue is worth $1 you would want to choose the one that will give you the most in the long run which would be red. The weighted probability and expected values all have a factor that affects the values of the outcomes.
Examples of these would be the lottery. The lottery has different values. Lower values have more of a chance of popping up than bigger values. And even though I’m against lottery because everything is sooo fake, that’s how it would work if they actually conducted the lottery fairly.
EVALUATION:
To solve this problem the main habit of a mathematician that I used was breaking it down. At first there was information just directly given and a lot of scenarios you had to solve but taking it part by part and adding it all together in the end really did help. I actually did really enjoy working on it and took a lot from it since I haven’t reviewed probabilities in this way in such a long time. There isn’t anything I would change because the problem was just simple and well brought together. Everything connected right and it wasn’t made up of just busy work. It kinda was pretty easy but that’s just until you actually understand it because the problem does make you think.
EXTENSION:
Archibald and Beatrice play a game in which they draw a card from a deck. Without taking in consideration of who draws the card:
- Jack: Beatrice pays $0.20 to Archibald
- Heart: Archibald pays Beatrice $0.08
- Neither Jack nor Heart: Both pay $0.01 to charity
After conducting an actual experiment (25 tests) our results came to be:
- 1 Jack is pulled
- Archibald gets money
- 5 Hearts are pulled
- Beatrice gets money
- 19 Non-Jacks/Hearts are pulled
- Charity gets money
- P(Archibald)=1/25(0.20)=0.008
- P(Beatrice)=5/25(0.08)=0.016
- P(Charity)=19/25(0.01)=0.0076
- Archibald : $0.008
- Beatrice : $0.016
- Charity : $0.0076