Bag of Fortune!

How the Game Would be Introduced

--> Say I had a class of 25 students, I would arrange the class into groups of five. At each station, I would have a sack filled with the same amount of marbles. The number of different colored marbles in each sack however would vary, so that the probability of choosing a specific color was different in each sack.

I would begin the game by having students start with one large sack at the front of the room. I would tell the students, "Inside of this bag, there are 40 marbles: 10 green, 10 blue, 10 red and 10 yellow, what color has the highest probability of being picked?"

I would then have students explain their answers and then explain the correct answer, that each color has the same probability of being picked because there are ten marbles of each color so the probabilities of the marbles inside of the bag are equally likely.

After this, I would explain that this is not the case in situations where the numbers of each color vary because then the probabilites of each outcome are not equally likely.

--> To emphasize the concept of solving probabilities that are not equally likely, I would then explain to the students the game we were going to play with the five different sack of marbles and game cards.

Rules of the Game:

--> Each student gets a game card

--> Students will be split into groups of five and each group will be assigned which sack to start at. Each sack will have a paper beside it on the desk telling students the number of blue, red, green, and yellow marbles inside.

-->The first member of the group will pull a marble. The group will then work together to solve the probability member one had of pulling that marble from the bag. Once reaching an agreement, member one will write the probability of their draw underneath the corresponding color on the game card and replace the marble.

-->The second, third, fourth and fifth member will then pull a marble, solve for the probability of their pull, write their probability down, and then replace the marble before the next person draws just as the first member did.

-->Each student will draw twice at each station and then rotate to the four remaining stations, so students will have a total of ten probabilities written down by the end of the game on their game card.

-->After finishing up at the five stations, the teacher will sit the students down and ask Group one at Sack One if each member in their group had the same probability of pulling a blue marble. So, did member two have the same probability of pulling a blue marble as member one did, even though member one pulled a marble out before member two did? Students would then discuss and explain their reasonings, leading into the discussion of independent events and replacement.

-->The teacher would segway into why each member of the group had the same probability as the other members of the group by explaining that because each member put the marble they pulled back, the pull of the member's before them in no way influenced their pull. Replacement allows the event of member one to not influence the event of member two because both member one and member two will pull from the exact same bag of marbles.

--> As a way to make the game fun and competitive, the teacher will tell the students to add up all ten of their probabilities (this will also refresh their addition and multiplication skills with fractions). Although we typically reward the person with the highest probability, so the least amount of difference between the numerator and denominator, this game would have a little twist. Instead, the teacher would say that the student who had the greatest difference between the numerator and denominator would be the winner because they pulled the marbles least likely to be pulled, the marbles with the lowest probabilities of being pulled.

Reflection of "Bag of Fortune":

--> Bag of Fortune would be a great way to introduce students to solving for probabilities that are NOT equally likely.

-->Bag of Fortune allows students to visually see and manipulate how the concept of replacement works. In this game, students pull out a marble but then replace it, allowing students to more easily see why the pull of member one does not affect the pull of member two, three, four, or five.

When a teacher first explains the concept of replacement, it could be confusing to a student how the probability could be the same for all the members even if members before had pulled out a marble, they may feel that this should change the probabilities/outcomes. When students are able to visually see member one put the marble they pulled back, it makes much more sense as to why member ones pull is independent from member twos pull, because the number of marbles stays the exact same throughout each members pulls.

--> This game is a great segway into the discussion of replacement and independent events.

--> This is a very interactive way for students to become engaged with the introduction of a new concept- solving for the probability of events that do not have equally likely probabilities and the introduction of replacement and independent events.

--> This game also allows students to refresh their skills on adding together fractions and finding common denominators when they add together their ten probabilities.