What are the chances???

Annenberg

Ah, probability. The first thing that comes to mind then I hear this word are topics like the weather and the lottery. I lived in Las Vegas, Nevada and worked in a casino. Everyday I was witness to people hoping that they would beat the odds and win the jackpot. When many people think of probability, they think of rolling dice, picking numbers at random, or playing the lottery. In fact, games of chance, which often involve dice or other random devices, rely on the principles of probability.

Working through the site I  found the problem very similar to what was covered in my statistics class a couple of years ago. 

I did come across a couple of problems that I just did not seem to get. 

Problem C8
Write a formula for determining the number of possible outcomes of n tosses of a fair coin. 

I think this confused me because I still struggle with algebraic equations.
Can you write out the steps for me so I can see it and walk through it myself? 

Problem C10
Use the binomial probability model to determine the following:
a. What is the most probable score you'll get?
b. What are the least probable scores you'll get?
c. What is the probability of getting at least two answers correct?
d. What is the probability of getting at least three answers correct?

I just could not think of how to start the problem. Is it just like the coin toss example?

A Whale of a Tale article 

Impossible	Certain	Likely	Unlikely
It is impossible for humans to breath in space.	It is certain that you will be another day older tomorrow.	It is likely that I will not get to bed on time any time soon.	It is unlikely that I will get another pet.
It is impossible to run faster than the speed of light.	It is certain that the sun will shin in the morning.	It is likely that I will go to school tomorrow.	It is unlikely that I will follow through with my New Years resolution.

Dice Toss 

Ms. Kincaid wanted the students to make predictions about their experiment on the basis of mathematical probability. Discuss preconceptions that students exhibited about tossing dice even after discussing the mathematical probability. Discuss the instructional implications of dealing with these preconceptions.

While introducing the lesson to her students Ms. Kincaid reminded her student about what they already knew about probability and if they could recall what they have learned previously about it and one student could recall an activity they did that evolved a coin toss.  Even though students seemed to know the definitions and could recall previous lessons, I still feel like they would benefit from more practice.  One student said she thought a two would come up more that 12 because in the past she had rolled a lot of ones.  However, the majority of the class felt that the number seven would come up a lot because there are more ways to make a seven using two dice than any other number.

Were these students too young to discuss mathematical probability? What evidence did you observe that leads you to believe that students did or did not grasp the difference between mathematical probability and experimental probability?  At what age should probability be discussed?

When asked students were able to recall the definitions to mathematical probability and experimental probability students were able to give appropriate responses. I do feel that they did grasp the material being taught.  I agree with Ms. Kincaid, we should introduce our younger students to probability so that they can develop their skills over time.  We should introduce our students to forms of probability in Kindergarten.  The simple basics such as, “If I drop this rock into water what is the probability it will sink or float?” to begin an introduction of vocabulary and modeled demonstration.

The teacher asked the students, “What can you say about the data we collected as a group?” and “What can you say mathematically?”  How did the phrasing of these two questions affect the students’ reasoning?

When she asked, “What can you say about the data we collected as a group?”, one student said that the data looked like a rocket.  This child told her what the data “looked” like to him.  When Ms. Kincaid asked the same child what it says mathematically it made him think about the data in a different way, he was able to tell her there were more sevens and gave her the reasoning behind his thought.

I feel that the way we word things particularly to children is very important.  Do you feel that you can get totally different answers to the same question just by asking the question in a different way?  

Why did Ms. Kincaid require each group of students to roll the dice thirty-six times?  What are the advantages and disadvantages of rolling this number of times?

6x6=36  this number is a fair amount to roll the dice because it give each number 1-6 a fair chance to be landed upon.  He down side of this is it may be hard to keep track of the number of rolls and students may over roll like the group in the video.

Comment on the collaboration among the students as they conducted the experiment.  Give evidence that students either worked together as a group or worked as individuals.

In each group, students were devising plans to implement their experiment with the dice.  One child discussed what had worked for him in a past group.  I feel the kids worked well together.  When one group realized that they had rolled to many times, they worked together to try to fix their data.

Why do you think Ms. Kincaid assigned roles to each group member?  What effect did this practice have on the students?  
How does assigning roles facilitate collaboration among the group members?

By assigning different roles to each member, it gives students equal responsibility for the work to be done in their group.  Everyone has to pay attention and work together in order for the data to be correctly gathered, recorded and analyzed.

Describe the types of questions that Ms. Kincaid asked the students in the individual groups. How did this questioning further student understanding and learning?

She asked her students if their results of the experiment were the same as their predictions.  She also asked them what turned out different than they thought.  She asks questions to gain insight into their thought process and to learn how they are arriving at their solutions.

Why did Ms. Kincaid let each group decide how to record the data rather than giving groups a recording sheet that was already organized?  When would it be appropriate to give students an organized recording sheet?  Discuss the advantages and disadvantages of allowing students to create their own recording plans.

She let her student pick their own way to organize their data to see if they as a group could chose a way to show the data clearly.  This allowed her to walk around the groups discussing their choices.  One group we saw mad a poor choice and had to reconfigure the data again so that her group could analyze it correctly. By allowing student to choose their own way of recording data, it allows them to visualize the goal and plan for the outcome of the interpretation.  This in turn gets students to critically think about what they are trying to accomplish and plan for their predicted outcome.  It would appropriate to give students an organized recording sheet during assessment of lesson.

For further consideration…

Knowing what you now know about probability concepts in the elementary school, how will you ensure that your students have the background to be successful with these concepts in the middle school?

I feel it is important to introduce young minds to probability concepts.  Students of all ages are touched by probability everyday in one way or another.  I feel it is very plausible to incorporate probability into activities in the classroom such as Science.  Connecting probability to concepts they already know and use will help with understanding.

Everyone loves Bill Nye!