Probability and Area

Today’s Learning Goals  To continue developing a deep understanding of theoretical probabilities.  We will begin to see probabilities related to area.  We will continue to develop a good understanding of the link between part-whole ratios, decimals, and percents.  We will continue to develop the ideas of equally likely and non-equally likely outcomes.

Review of Probability Previously, we discussed introductory ideas of probability. For example, suppose the spinner at the right was spun. What is the probability that the spinner would land on yellow? Good…¼ or 25%. What is the probability that the spinner would land on green? Correct…¼ or 25%.

Probability and Area Mr. Jackson gave his class the following spinner. Marcus said that the probability the spinner would land on yellow was ¼ or 25% because yellow was 1 out of 4 parts. Is Marcus correct? No…although yellow is 1 out of four parts, the parts are not equal! Every probability problem we have discussed to this point had equal parts!

Probability and Area If the red and green parts are equal-sized, what is the minimum number of pieces the circle could be cut into so that everything has the same-sized pieces? Nice…cut the circle up into eight equal parts. Now that everything is cut up into equal-sized parts, what is the probability of getting yellow? Beautiful… = 50%.

Probability and Area The probability the spinner would land on yellow is 4/8 or 50%. We can see that yellow is half of the circle. Again, you can see that blue is ¼ of the circle. So, it has less of a chance of getting picked than yellow. Great… = 25%. What is the probability that the spinner would land on blue? What is the probability that the spinner would land on green? Excellent… = 12.5%.

Probability and Area So, we have the following probabilities: What is true about the sum of all of these probabilities? P(yellow) = ½ = 50% P(blue) = ¼ = 25% P(green) = 1/8 = 12.5% P(red) = 1/8 = 12.5% Wonderful…they all add up to 1 whole or 100%.

Probability and Area Today, we will explore a computer game called Treasure Hunt. The first level of the Treasure Hunt game is shown at the right. The computer hides a treasure in one of the rooms. It is your job to figure out which room the treasure is in with the least amount of guesses.

Level 1 of Treasure Hunt The computer gives clues about where the treasure is located. After each clue, the player must guess which room the treasure is in. Suppose the computer gives you this first clue: The treasure is hidden in a room with “hall” in its name. Which room should you guess first? Explain your answer. Yes…the Great Hall because this room appears to be bigger than the dining hall!

Level 1 for Treasure Hunt For example, if the computer selected the square indicated on the grid at the right, the treasure was hidden in the conservatory. To make good guesses when playing Treasure Hunt, it helps to understand how the computer hides the treasure. The computer “thinks” of the first floor of the palace as a 10 by 10 grid. The computer randomly selects one of the 100 squares as the location for the treasure.

Level 1 for Treasure Hunt Awesome…16/100 = 16%. Now that you see the equal-sized parts, what is the probability that the treasure would be hidden in the Dining Hall? What is the probability that the treasure would be hidden in the Great Hall? Fantastic…30/100 = 30%. So, based on the probabilities, which of these two rooms is the computer more likely to put the treasure in? Super…the Great Hall.

Level 1 for Treasure Hunt No…because the servant’s chamber is 4 out of 100 squares and the entrance corridor is 6 out of 100. So, there is more of a chance that the treasure will be put in the entrance corridor. Monty says that since the servant’s chamber and entrance corridor are both small rooms, they have the same chance of having the treasure. Is Monty correct? Explain.

Level 1 of Treasure Hunt Is it equally likely that the treasure will be hidden in any of the 100 squares in the grid? Explain why or why not. Excellent…each one is equally likely because they are all the same size and the computer RANDOMLY puts the treasure into one of them. Is it equally likely that the treasure will be hidden in any of the rooms? Explain why or why not. Good…each room is not equally likely to be chosen because they are NOT all the same size.

Partner Work You have 20 minutes to work on the following problems with your partner.

For those that finish early In games like Treasure Hunt in which probabilities are related to area, how can you tell if two events are equally likely?

Big Idea from Today’s Lesson To determine the theoretical probabilities of something occurring that involves the area of an object, cut it up into equal parts so that each part is equally likely to all other parts.

Homework Complete Homework Worksheet.