what are the chances?

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**Probability portfolio**

This year in math, the first big unit of the year was based on probability and chance. The main idea of studying this concept is to be able to use the tool of probability to estimate and guess the outcome of many different situations. Whether you do this with simulations and mathematical experiments, experimental probability, or you use calculations, area diagrams, and “common sense” to predict the outcome, theoretical probability, you can always use the idea of probability to explore and predict the outcomes to many “Games” of the real world.

Throughout this unit we solved many problems that used both types of probability and problems that were more focused on one of the types of probability. An example of a lesson where we used both types of probability was the Spinner Give and Take lesson. The premise for this game was as follows: Betty and Al are playing a game where they have a spinner split into four pieces. If the arrow lands on three of the four Al gives Betty $1 and if it lands on the only other spot Betty gives Al $4. Using this game we used both types of probability to see whether Betty or Al won a game, the way we used theoretical probability was by calculating the chances each person had and using those chances to predict the outcomes of the game to see who had the overall profit. We started the problem by finding that Betty had a ¾ chance of having the arrow land on her and that Al had a ¼ chance. Using this information we decided to have a game of 25 turns, of those 25 the arrow landed on Betty 19 times and on Al 6 times. But we found that in that game Betty made $19 while Al made $24. Finding that answer we concluded that theoretically Al should win the game. The way we tested this game experimentally was by simulating our own game with a spinner split in four. We also played a game of 25 turns and of those 25 the arrow landed on Betty 19 times while only landed on Al 6 times giving the same result as the theoretical probability, proving that Al indeed won the game.

Another example of a lesson we solved that exclusively used the theoretical aspect of this concept was the Gum ball Dilemma. For this problem we had to find a formula to fit a situation where you can find the amount of money, in pennies, a parent must spend if her kids want gum balls of the same color from a gumball machine in which "X" amount of gum balls cost one penny per gum ball. For this problem you had to give multiple situations where the parent had a certain amount of kids and the gum ball machine had a certain amount of gum ball colors. With that information you had to find equations that fit the situation and predict how much money the parent was going to spend after analyzing the pattern that the gum balls had due to their probability of colors. This was by far the most in depth lesson we had and probably the one I am most proud of. For a more detailed look at the lesson please visit the second Problem of the Week page under 10th Grade Math.

Two of the Habits of a Mathematician I used most in this unit were “Conjecture and Test” and “Seek Why and Prove”. I used these very often and in a connected way for the both types of probability we were working with. I used Conjecture and Test in theoretical probability by using it to present a situation and using the information given to seek multiple solutions to the situation by working and proving it through the testing. The same relationship presented itself with experimental probability by having me seeking how the experiment was true and proving it by conjecturing it and testing it in multiple ways.

Throughout this unit we solved many problems that used both types of probability and problems that were more focused on one of the types of probability. An example of a lesson where we used both types of probability was the Spinner Give and Take lesson. The premise for this game was as follows: Betty and Al are playing a game where they have a spinner split into four pieces. If the arrow lands on three of the four Al gives Betty $1 and if it lands on the only other spot Betty gives Al $4. Using this game we used both types of probability to see whether Betty or Al won a game, the way we used theoretical probability was by calculating the chances each person had and using those chances to predict the outcomes of the game to see who had the overall profit. We started the problem by finding that Betty had a ¾ chance of having the arrow land on her and that Al had a ¼ chance. Using this information we decided to have a game of 25 turns, of those 25 the arrow landed on Betty 19 times and on Al 6 times. But we found that in that game Betty made $19 while Al made $24. Finding that answer we concluded that theoretically Al should win the game. The way we tested this game experimentally was by simulating our own game with a spinner split in four. We also played a game of 25 turns and of those 25 the arrow landed on Betty 19 times while only landed on Al 6 times giving the same result as the theoretical probability, proving that Al indeed won the game.

Another example of a lesson we solved that exclusively used the theoretical aspect of this concept was the Gum ball Dilemma. For this problem we had to find a formula to fit a situation where you can find the amount of money, in pennies, a parent must spend if her kids want gum balls of the same color from a gumball machine in which "X" amount of gum balls cost one penny per gum ball. For this problem you had to give multiple situations where the parent had a certain amount of kids and the gum ball machine had a certain amount of gum ball colors. With that information you had to find equations that fit the situation and predict how much money the parent was going to spend after analyzing the pattern that the gum balls had due to their probability of colors. This was by far the most in depth lesson we had and probably the one I am most proud of. For a more detailed look at the lesson please visit the second Problem of the Week page under 10th Grade Math.

Two of the Habits of a Mathematician I used most in this unit were “Conjecture and Test” and “Seek Why and Prove”. I used these very often and in a connected way for the both types of probability we were working with. I used Conjecture and Test in theoretical probability by using it to present a situation and using the information given to seek multiple solutions to the situation by working and proving it through the testing. The same relationship presented itself with experimental probability by having me seeking how the experiment was true and proving it by conjecturing it and testing it in multiple ways.

__Some examples of my work:__

- The Pow #2 Page under the Problem of the week page