Math Focus 9
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# Probability in Life

## INTRODUCTION

Decisions made on the basis of probability may include a combination of theoretical probability, experimental probability, and subjective judgment. When using probability, it is important to recognize the difference between cause and coincidence. Coincidence can be considered something that happens by chance.

Your task is to conduct virtual experiments and relate the results to decision making. You can choose between the following two experiments or conduct both. Then you will discuss a probability question with someone and make a few notes about your discussion.

• Research the probability of precipitation in weather forecasts, and explain whether the forecast would influence your decision. Then determine the probability of precipitation in different ways, including experiments with virtual spinners and dice, and revisit your decision.
• Use virtual experiments to explore the probability of birthdays on the same day. You will decide whether you agree with a probability statement, and you will make your own probability statement about birthdays in your birthday month.

## THE PROCESS

Steps 1 to 5 are for the precipitation experiment. Steps 6 to 9 are for the birthday experiment. Step 10 is a discussion question about probability.

Precipitation Experiments

1. Go to the Environment Canada Weatheroffice. Find a location where there is a probability of precipitation for at least two of the days in the forecast. Record the location, the days in the forecast (not including today), and the probability of precipitation in each. Why does it make sense to record 0% probability for a day when precipitation is not mentioned?
2. Choose one of the following events, or make up an outdoor event that would take place on each of the days in the forecast. Decide whether the probability of precipitation would influence your plans for the event.
• Go on a camping trip that is appropriate for the season and location.
• Have an outdoor fair.
• Go shopping at an outdoor market each day.
• Plan a series of concerts or plays that would last all day and evening.
1. Go to Spinners. Change the spinner so that the number of sections of one colour models the percent of the probability for precipitation on the first day. Spin the spinner. How does the result relate to precipitation on that day? Continue for each day in the forecast. Do you need to spin the spinner when the forecast does not mention precipitation? Repeat the experiment at least five times. How does this experiment model the probability of precipitation?
2. Go to Virtual Dice. Click on Customize Your Die and select the die with 10 faces. Type words for the faces to model the percent of the precipitation for the first day. For example, you might type yes, yes, no, yes, no, yes, no, yes, no, yes to represent a 60% chance of precipitation, or type rain, rain, rain, rain, none, none, none, none, none, none to represent 40% chance of rain. What could you type if the forecast does not show a percent of precipitation? Click on Create, and then toss the die. How does the result relate to rain that day? Continue for each day in the forecast. Repeat the experiment at least five times. How does this experiment model the probability of precipitation?
3. Would the results of your experiments affect your decision for step 2? What assumptions did you make? Are the assumptions reasonable? Do you think that it matters that the events might only take place during the day?

Birthday Experiments

1. Go to Birthday Coincidence and read that in a group of 23 people, the probability of at least two having the same birthday is greater than ½, or 50%. Explain this in your own words.
2. Go to Random Number Generator and Checker. Enter the numbers 23, 1, and 365 to have it generate 23 integers between 1 and 365. Click GO. Record the result with one tally in a tally chart such as the following.
 Some Numbers the Same No Numbers the Same /

Repeat this at least 20 times, marking one tally for each trial. How does this experiment model the Birthday Coincidence? Do you agree with the Birthday Coincidence? Explain.

1. Suppose there is a group of people whose birthday is in the same month as yours. How many do you think need to be in the group for someone to have the same birthday as you 50% of the time? Go to Random Integer Generator. For “Generate _ random integers,” enter the number of people you think need to be in the group. For “Each integer should have a value between _1_ and _100_,” enter 1 in the first box and the number of days in your birthday month in the second box. Click Get Numbers. Record the result with one tally in a tally chart such as the following.
 Same Day as My Birthday Not Same Day as My Birthday

Repeat this 10 times, marking one tally for each trial. How does this experiment model the number of people needed in the group?

1. Change the number of people needed in the group for step 8, and repeat the experiment until you can draw a conclusion about the number of people required in the group. Make a statement about the number of people needed in the group. How do your experiments support your statement?

Discussion

1. Choose one of these questions to discuss with someone. Then write a few notes about your discussion.
• Why is it important to conduct an experiment several times?
• How have you used probability in your own life? Could an experiment be helpful?
• How do you think you might use probability in your own life? Would you use an experiment?
• What are advantages and disadvantages of thinking of probability when making a decision?

## RESOURCES

Websites:

Spinners

Virtual Dice

Birthday Coincidence

Random Number Generator and Checker

Random Integer Generator

Materials:

a pencil
paper

## ASSESSMENT

 Mathematical Processes Work meets standard of excellence Work meets standard of proficiency Work meets acceptable standard Work does not yet meet acceptable standard Communication Provides a precise and insightful explanation of mathematical concepts and the simulations. Provides a clear and logical explanation of mathematical concepts and the simulations. Provides a partially clear explanation of mathematical concepts and the simulations. Provides a vague and/or inaccurate explanation of mathematical concepts and the simulations. Connections Makes insightful connections among the virtual simulations, the situations, and probability. Makes meaningful connections among the virtual simulations, the situations, and probability. Makes simple connections among the virtual simulations, the situations, and probability. Makes minimal or weak connections among the virtual simulations, the situations, and probability. Reasoning Comprehensively analyzes the experiments strategies and results to make insightful generalizations. Completely analyzes the experiments strategies and results to make logical generalizations. Superficially analyzes the experiments strategies and results to make simple generalizations. Is unable to analyze the experiments strategies and results to make generalizations.