Chapter 5 Project Coin Toss Game

by Pietro Cameli                                         24/5/2015

Introduction

This project is about proving if games are fair or unfair using Probability calculations and Diagrams.

Game Number 1

In this game two players toss 3 coins. If the result is exactly 2 heads, Player One wins. If the result is anything else, Player Two wins.

To prove that the game is fair or unfair, you need to calculate the probability of winning for player 1 and 2. These two events are mutually exclusive and complementary, because if Player One wins, Player Two can't win and if Player Two wins, Player One can't win (mutually exclusive) and the probability of Player One and Two added together are a whole (1) (Complementary). So you just have to calculate the probability of winning of one of the 2 players and subtracting it from the whole (1) to have the probability of winning of the other player. The game is fair if the 2 probabilities are equal (1/2 each).

The Calculation of the probabilities

I made a diagram (Three Coin Toss Game Diagram) to show the Total Possible Outcomes of the three coins and counted how many there were: 8.

Then, I counted the Possible Winning Outcomes for Player One: 3 and I calculated the Probability of Winning of Player One

Possible Winning Outcomes for Player One/Total Possible Outcomes = 3/8

Then I calculated the Probability of Winning of Player Two, subtracting the Probability of Winning of Player One from the Whole (8/8), because the events are Complementary

8/8 - 3/8 = 5/8

and I compared them

3/8 < 5/8.

The Venn Diagram

This is the Venn diagram: the two sets of outcomes are disjoint: there are no outcomes that are tie.

The results

The game isn't fair because Player 2 has 5/8 chance of winning and Player 1 has only 3/8.

Game Number 2

In this game there are 2 players one against the other with 12 cards: the Ones , Twos, and Threes of all the types of symbols: Hearts, Spades, Clubs, Diamonds. Each player can pick one card from the deck and wins against the other player if he/she has a card that has the largest value or if he/she picks a Hearts. If the two players pick a Hearts at the same time, the one with the largest value wins.

Below you can see me playing with my dad the game.

The Calculation of the probabilities

I made a diagram (The Luck Card Game Diagram) using an  Excel table to show all the Possible Outcomes of the game, to understand if the game is fair or not. I counted how many outcomes there were: 132. Then I counted the Winnings Outcomes for Player One and Player Two: for both here are 57 Winnings Outcomes and there are 18 Outcomes that are tie. So the events that Players One and Two wins are not Complementary because there are Outcomes which are tie,and so the probability that Player One and Player Two added together are not a Whole. I made a fraction to represent the probability of winnings of the two player Possible Winning Outcomes of a player/ Total Possible Outcomes = 57/132

and a fraction to represent the probability of draw

Possible Outcomes of draw/ Total Possible Outcomes =

18/132=3/22

The Venn Diagram

In the Venn diagram below you see that there are two overlapping set representing the possible outcomes, the overlapping part show the outcomes related to a tie for the two players. The event that Player one or Two win are mutually exclusive, because if one wins the other cannot. They are not complementary because if added together the result is not a Whole, because of the outcomes of draw

The Result

Looking at all the possible outcomes all the players have the same amount of probability of winning 43% each.

57/132*100=43% rounded to the ten place

In 14% of the cases the game is a draw.

3/22*100=14% rounded to the ten place

So according to the probability we can say the game is fair.

Game Number 3

In this game there are 2 players one against the other with 14 cards: 1, 2, 3, 4, and King are red card, 5,6,7,8,9,10, Jack and Queen are black card and a Jolly. Player One can wins when he picks up the Jolly or a black card while Player Two has a black card, Player Two wins when he picks up the Jolly or a red card while Player One has a red card. In all the other cases is a draw.

In the picture below my mom wins, because she has Hearts

The calculation of the probabilities

Like in Game Number 2 I made a diagram (The color Card Game Diagram) Excel table to show all the possible outcomes of the game, to understand if the game is fair or not. I counted how many outcomes there were: 182. Then I counted the Winnings Outcomes for Player One (33) and Player Two (69): for player 1 there are 80 outcomes which are tie. I made a fraction to represent the probability of winnings of the two players:

Possible Winning Outcomes Player One/ Possible Total Outcomes = 33/182

Possible Winning Outcomes Player Two/ Possible Total Outcomes = 69/182

and a fraction to represent the probability of draw

Possible draw/ Possible Total Outcomes = 80/182 = 40/91

Venn diagram

In the Venn diagram below you see that there are two overlapping set representing the possible outcomes. The overlapping part show the outcomes that are a tie for the two players. The event that Player One or Two win are mutually exclusive, because if one wins the other cannot. They are not complementary because if added together the result is not a Whole, because of the tied outcomes.

The results

Looking at all the possible outcomes all the players have a different amount of probability of winning: 18% for Player One

33/182*100 = 18% rounded to the Ones place

and 38% for Player Two

69/182*100 = 38% rounded to the `One Place

44% of the game is a draw

40/91*100 = 44%.

So according to the probability we can say the game is unfair because

18%<38%.

Conclusion

The project was really fun trying to find the probability of winning and playing the games and testing how they worked. I think this was the funniest project of the year and I could practice how to calculate quickly the chances that an event can happen. This will help to make better evaluations in my daily life (not to mention now I can win at the casino or open my own!).

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