May the odds be ever in your favor
"The students will study probability by looking at the possible outcome of various games. They will then design and build their own original casino-style game in the hope of tempting would-be gamblers to lose their money and make them rich in the process. The game must be fair enough to tempt bets, but the odds must also be stacked sufficiently in your favor to make (not lose) you money." - Carlee Hollenbeck
group game
Lightyear
Description
My group's casino game is called LIGHTYEAR. The game is very similar to pinball as there is a slanted board and the player needs to shoot a marble from one end of the board to the other. On the side of the board opposite the sling shot, there is a circular formation of targets. There are three categories of different sized targets, ranging from seven trapezoids to six circles to one smaller circle (the jackpot). Each of the shapes is a different color, either green, red, or yellow. Each color is worth a different value. Red is a loss for the opponent and they have to pay 100. Yellow is neutral and they continue to their next turn. Green is a win for the opponent.
Rules
- Pay a flat rate of $200
- 3 turns each
- WIN: 5 → 3 ~Trapezoids + 2 Circles = every win slot you get $100
- LOSE: 6 → 3 ~Trapezoids + 3 Circles = $0 and can't try again
- NEUTRAL: 3 → 2 ~Trapezoids + 1 Circle = $0 and try again
- JACKPOT: $250
- If the doesn't go into the slots → retry
Odds
Diameter of Total Wheel: 144 cm
Radius: 72 cm
Area: 3.14(72)2 = 16,277.8 cm2
Total Area: 16,277.8 cm2
Diameter of Medium Circles: 12 cm
Radius: 6 cm
Area: 3.14(6)2 = 113.04 cm2
Total for all circles: 113x6 = 678cm2
Diameter of Jackpot Circle: 10 cm
Radius: 5 cm
Area: 3.14(5)2 = 78.5 cm2
Total Area: 78.5 cm2
Trapezoid Dimensions: 27cmx20cmx42cm
Area: ½ h(b1+b2)
Area: 10(69) = 690 cm2
Total for all trapezoids: 690x8 = 5,520 cm2
Probability of Small Circle: 78.5cm2 16,277.8cm2 = 796216,278 = 0.48%
Probability of Red Spots (6)
Trapezoids (4) = 690cm2x4 = 2,760
Circles (2) = 113cm2x2 = 226
Percentage: 2,98616,278 = 18%
Probability of Green Spots (5)
Trapezoids (2) = 690cm2x2 = 1,380
Circles (3) = 113cm2x3 = 339
Percentage: 1,719 16,278 = 10.5 %
Probability Yellow (3)
Trapezoids (2) = 690cm2x2 = 1,380
Circles (1) 113cm2x1 = 113
Percentage: 1,493 16,278 = 9.1%
Non-Slotted Space (Rest of the Wheel)
Trapezoids (8) = 5,520
All Circles = 756.5
Total: 6,276.5
Percentage: 6,276.5 16,278 = 38.5%
Reflection
To make the studying of odds more exciting and playful, we can study the odds of getting winning combinations in various casino games. To strategize on games that appear to be luck, we simply break down the winning combinations and use probability rules to calculate the odds of a certain player attaining those combinations based on the cards, dice, etc. that they are holding. To display difficult mathematic content to others in a simple way, we can use diagrams and writing. A diagram provides a visual for a pupil to examine and understand how different variables effect certain outcomes. By writing step by step analysis, it is easy for one to observe the process of calculating probability. I felt that I was most successful on the group aspect of this project. I had a wonderful time coming up with our game board idea/design and building it with my amazing partners, Charlotte and Sydney. I clearly understood the math that was behind calculating the probability of each combination on our game board and feel confident that I can explain it to others.
My group's casino game is called LIGHTYEAR. The game is very similar to pinball as there is a slanted board and the player needs to shoot a marble from one end of the board to the other. On the side of the board opposite the sling shot, there is a circular formation of targets. There are three categories of different sized targets, ranging from seven trapezoids to six circles to one smaller circle (the jackpot). Each of the shapes is a different color, either green, red, or yellow. Each color is worth a different value. Red is a loss for the opponent and they have to pay 100. Yellow is neutral and they continue to their next turn. Green is a win for the opponent.
Rules
- Pay a flat rate of $200
- 3 turns each
- WIN: 5 → 3 ~Trapezoids + 2 Circles = every win slot you get $100
- LOSE: 6 → 3 ~Trapezoids + 3 Circles = $0 and can't try again
- NEUTRAL: 3 → 2 ~Trapezoids + 1 Circle = $0 and try again
- JACKPOT: $250
- If the doesn't go into the slots → retry
Odds
Diameter of Total Wheel: 144 cm
Radius: 72 cm
Area: 3.14(72)2 = 16,277.8 cm2
Total Area: 16,277.8 cm2
Diameter of Medium Circles: 12 cm
Radius: 6 cm
Area: 3.14(6)2 = 113.04 cm2
Total for all circles: 113x6 = 678cm2
Diameter of Jackpot Circle: 10 cm
Radius: 5 cm
Area: 3.14(5)2 = 78.5 cm2
Total Area: 78.5 cm2
Trapezoid Dimensions: 27cmx20cmx42cm
Area: ½ h(b1+b2)
Area: 10(69) = 690 cm2
Total for all trapezoids: 690x8 = 5,520 cm2
Probability of Small Circle: 78.5cm2 16,277.8cm2 = 796216,278 = 0.48%
Probability of Red Spots (6)
Trapezoids (4) = 690cm2x4 = 2,760
Circles (2) = 113cm2x2 = 226
Percentage: 2,98616,278 = 18%
Probability of Green Spots (5)
Trapezoids (2) = 690cm2x2 = 1,380
Circles (3) = 113cm2x3 = 339
Percentage: 1,719 16,278 = 10.5 %
Probability Yellow (3)
Trapezoids (2) = 690cm2x2 = 1,380
Circles (1) 113cm2x1 = 113
Percentage: 1,493 16,278 = 9.1%
Non-Slotted Space (Rest of the Wheel)
Trapezoids (8) = 5,520
All Circles = 756.5
Total: 6,276.5
Percentage: 6,276.5 16,278 = 38.5%
Reflection
To make the studying of odds more exciting and playful, we can study the odds of getting winning combinations in various casino games. To strategize on games that appear to be luck, we simply break down the winning combinations and use probability rules to calculate the odds of a certain player attaining those combinations based on the cards, dice, etc. that they are holding. To display difficult mathematic content to others in a simple way, we can use diagrams and writing. A diagram provides a visual for a pupil to examine and understand how different variables effect certain outcomes. By writing step by step analysis, it is easy for one to observe the process of calculating probability. I felt that I was most successful on the group aspect of this project. I had a wonderful time coming up with our game board idea/design and building it with my amazing partners, Charlotte and Sydney. I clearly understood the math that was behind calculating the probability of each combination on our game board and feel confident that I can explain it to others.
Individual analysis
go fish
Description of Game (What is the game and how do you play?)
Each player draws five cards.
Randomly choose a player to go first.
On your turn, ask a player for a specific card rank. You must already hold at least one card of the requested rank.
If the player you ask has any cards of the requested rank, she must give all of her cards of that rank to you.
If you get one or more cards from the player you ask, you get another turn. You may ask any player for any rank you already hold, including the same one you just asked for.
If the person you ask has no relevant cards, they say, "Go fish." You then draw the top card from the draw pile.
If you happen to draw a card of the rank asked for, show it to the other players and you get another turn. However, if you draw a card that's not the rank you asked for, it becomes the next player's turn. You keep the drawn card, whatever rank it is.
When you collect a set of four cards of the same rank, you show the set to the other players and put the four cards face down on the table.
Winning Combinations (What has to happen for you to win?)
In order to win you must accumulate the highest number of piles. These piles consist of 4 matching cards.
Example: Sally and Roy are playing Go Fish. Sally has 6 piles of 4 matching cards in front of her at the end of the game. Roy has 7 piles of 4 matching cards. Because Roy has one more pile than Sally, he is the winner.
Probability of Each Combination (Explain the process of computing the probability.)
What is the probability of receiving a 2 on your first draw? (All of the 2’s are located in the draw pile, not in your opponents hand.)
There are 52 cards in a deck.
There are 4 cards for each value in the deck.
Therefore the probability is: 4/52 or 7%
You now have 4 piles of matched cards. Your opponent has 5 piles of matched cards. You are holding four cards in your hand and you have a 2 within those four cards. Your opponent is holding 5 cards and does not have a 2 within those 5. What is the probability that you will draw a 2 from the pile?
There are a total of 9 piles of matched cards on the table.
There are 4 cards within each pile.
There are 4 cards in your hand and 5 cards in your opponents hand. This renders another 9 cards out of play meaning that 7 cards are still in play.
There are now a total of 10 piles of matched cards on the table between you and your partner. You are holding five cards and your partner is holding four. It is their turn to draw. They are hoping to draw a 4 from the pile. There is one four in your hand. What is the probability that they will draw a four?
There were originally 52 cards in the deck. There are 10 piles on the table, each containing 4 cards.
Expected Outcome (Describe the expected outcome and likely of getting the winning combination.)
It is very unlikely that a player will receive a winning combination on the first draw. A winning combination is four of a kind and the probability of drawing four of the same kind initially, is very low. In order to get a winning combination a player must continue to draw and collect the cards that they desire. However, as the game goes on, there are less cards in play, but all of the cards in play are cards that the players desire, as they have not yet been collected into a group of four of the same. So the likelihood of a player finishing the game quickly and collecting their matches is very probable.
Each player draws five cards.
Randomly choose a player to go first.
On your turn, ask a player for a specific card rank. You must already hold at least one card of the requested rank.
If the player you ask has any cards of the requested rank, she must give all of her cards of that rank to you.
If you get one or more cards from the player you ask, you get another turn. You may ask any player for any rank you already hold, including the same one you just asked for.
If the person you ask has no relevant cards, they say, "Go fish." You then draw the top card from the draw pile.
If you happen to draw a card of the rank asked for, show it to the other players and you get another turn. However, if you draw a card that's not the rank you asked for, it becomes the next player's turn. You keep the drawn card, whatever rank it is.
When you collect a set of four cards of the same rank, you show the set to the other players and put the four cards face down on the table.
Winning Combinations (What has to happen for you to win?)
In order to win you must accumulate the highest number of piles. These piles consist of 4 matching cards.
Example: Sally and Roy are playing Go Fish. Sally has 6 piles of 4 matching cards in front of her at the end of the game. Roy has 7 piles of 4 matching cards. Because Roy has one more pile than Sally, he is the winner.
Probability of Each Combination (Explain the process of computing the probability.)
What is the probability of receiving a 2 on your first draw? (All of the 2’s are located in the draw pile, not in your opponents hand.)
There are 52 cards in a deck.
There are 4 cards for each value in the deck.
Therefore the probability is: 4/52 or 7%
You now have 4 piles of matched cards. Your opponent has 5 piles of matched cards. You are holding four cards in your hand and you have a 2 within those four cards. Your opponent is holding 5 cards and does not have a 2 within those 5. What is the probability that you will draw a 2 from the pile?
There are a total of 9 piles of matched cards on the table.
There are 4 cards within each pile.
- 9 x 4 = 36
- 52 - 36 = 16
There are 4 cards in your hand and 5 cards in your opponents hand. This renders another 9 cards out of play meaning that 7 cards are still in play.
- 5 + 4 = 9 and 16 - 9 = 7
- 3 / 7 or 42.8%
There are now a total of 10 piles of matched cards on the table between you and your partner. You are holding five cards and your partner is holding four. It is their turn to draw. They are hoping to draw a 4 from the pile. There is one four in your hand. What is the probability that they will draw a four?
There were originally 52 cards in the deck. There are 10 piles on the table, each containing 4 cards.
- 10 x 4 = 40
- 52 - 40 = 12
- 4 + 5 = 9
- 12 - 9 = 3
- 3/3
Expected Outcome (Describe the expected outcome and likely of getting the winning combination.)
It is very unlikely that a player will receive a winning combination on the first draw. A winning combination is four of a kind and the probability of drawing four of the same kind initially, is very low. In order to get a winning combination a player must continue to draw and collect the cards that they desire. However, as the game goes on, there are less cards in play, but all of the cards in play are cards that the players desire, as they have not yet been collected into a group of four of the same. So the likelihood of a player finishing the game quickly and collecting their matches is very probable.
crazy 8's
Description of Game (What is the game and how do you play?)
The player to the left of the dealer goes first and then clockwise.
On a turn, each player adds to the discard pile by playing one card that matches the top card on the discard pile by suit or rank.
A player who cannot match the top card on the discard pile by suit or rank must draw cards until he can play one. It is allowed to pull cards from the draw pile even if you already have a legal play. When the draw pile is empty, a player who cannot add to the discard pile passes his turn.
All eights are wild and can be played on any card during a player's turn. When a player discards an eight, he chooses which suit is now in play. The next player must play either a card of that suit or another eight.
Winning Combinations (What has to happen for you to win?)
The first player to discard all of his cards wins.
Probability of Each Combination (Explain the process of computing the probability.)
There are 20 cards in the discard pile and 16 cards in each players hand. There is a jack of spades on the top of the pile and there has already been one queen discarded. What is the probability that you or your partner is holding a queen?
There are 20 cards out of play.
52 - 20 = 32
32 / 2 = 16 (cards in each players hand)
One 3 has already been discarded.
4 - 1 = 3
There are 3 queens left in play.
3 / 16 = 0.187
19%
The probability of you or your partner holding a queen is 19%
There are 20 cards in the discard pile and 16 cards in each players hand. There is a jack of spades on the top of the pile and there has already been one queen discarded. What is the probability that you or your parter is holding a spade? (There have already been 5 spades played in the discard pile.)
There are 52 cards in a deck and there are 4 of each card. One card in each set of 4 is a spade.
52 / 4 = 13
This means that there are 13 spades in each deck. Five spades are no longer in play.
13 - 5 = 8
There are 8 spades in play between you and your partner.
8 / 16 = 2
There is a 50% chance that you or your partner is holding a spade.
There are now 30 cards in the discard pile and 11 cards in each players hand. There is a 4 of hearts on the top of the pile. What is the probability that you or your partner is holding a 5 that is also a heart? (There is one five and seven hearts in the discard pile.)
There are 30 cards in the discard pile
52 - 30 = 22
There are 22 cards between you and your opponent. There is one five in the discard pile.
4 - 1 = 3
There are 3 fives left in play. There are seven hearts in the discard pile.
13 - 7 = 6
There are 6 hearts left in play. To determine the probability, we must use the “Not Mutually Exclusive” rule. (The probability of the first outcome + probability of second outcome - the probability of both.)
3 / 22 + 6 / 22 - 1 / 22 = 8 / 22
36%
Expected Outcome (Describe the expected outcome and likelihood of getting the winning combination.)
The player to first discard all of their cards is the winner. The cards must be placed on the discard pile in a certain order. Either they must be the same number as the number on the top of the pile or the same suit as the suit on the top of the discard pile. Since a player can place cards by suit or rank, the probability of them holding the card that is needed is much more likely than it would be if it was only by rank, or only by suit, therefore it is very likely that a player will be holding the necessary cards.
The player to the left of the dealer goes first and then clockwise.
On a turn, each player adds to the discard pile by playing one card that matches the top card on the discard pile by suit or rank.
A player who cannot match the top card on the discard pile by suit or rank must draw cards until he can play one. It is allowed to pull cards from the draw pile even if you already have a legal play. When the draw pile is empty, a player who cannot add to the discard pile passes his turn.
All eights are wild and can be played on any card during a player's turn. When a player discards an eight, he chooses which suit is now in play. The next player must play either a card of that suit or another eight.
Winning Combinations (What has to happen for you to win?)
The first player to discard all of his cards wins.
Probability of Each Combination (Explain the process of computing the probability.)
There are 20 cards in the discard pile and 16 cards in each players hand. There is a jack of spades on the top of the pile and there has already been one queen discarded. What is the probability that you or your partner is holding a queen?
There are 20 cards out of play.
52 - 20 = 32
32 / 2 = 16 (cards in each players hand)
One 3 has already been discarded.
4 - 1 = 3
There are 3 queens left in play.
3 / 16 = 0.187
19%
The probability of you or your partner holding a queen is 19%
There are 20 cards in the discard pile and 16 cards in each players hand. There is a jack of spades on the top of the pile and there has already been one queen discarded. What is the probability that you or your parter is holding a spade? (There have already been 5 spades played in the discard pile.)
There are 52 cards in a deck and there are 4 of each card. One card in each set of 4 is a spade.
52 / 4 = 13
This means that there are 13 spades in each deck. Five spades are no longer in play.
13 - 5 = 8
There are 8 spades in play between you and your partner.
8 / 16 = 2
There is a 50% chance that you or your partner is holding a spade.
There are now 30 cards in the discard pile and 11 cards in each players hand. There is a 4 of hearts on the top of the pile. What is the probability that you or your partner is holding a 5 that is also a heart? (There is one five and seven hearts in the discard pile.)
There are 30 cards in the discard pile
52 - 30 = 22
There are 22 cards between you and your opponent. There is one five in the discard pile.
4 - 1 = 3
There are 3 fives left in play. There are seven hearts in the discard pile.
13 - 7 = 6
There are 6 hearts left in play. To determine the probability, we must use the “Not Mutually Exclusive” rule. (The probability of the first outcome + probability of second outcome - the probability of both.)
3 / 22 + 6 / 22 - 1 / 22 = 8 / 22
36%
Expected Outcome (Describe the expected outcome and likelihood of getting the winning combination.)
The player to first discard all of their cards is the winner. The cards must be placed on the discard pile in a certain order. Either they must be the same number as the number on the top of the pile or the same suit as the suit on the top of the discard pile. Since a player can place cards by suit or rank, the probability of them holding the card that is needed is much more likely than it would be if it was only by rank, or only by suit, therefore it is very likely that a player will be holding the necessary cards.