10-27. On the midway at the county fair, there are many popular games to play. One of them is "Flip to Spin or Roll." You start by flipping a coin. If heads comes up, you get to spin the big wheel, which has ten equal sectors: three red, three blue, and four yellow. If the coin shows tails, you get to roll a cube with three red sides, two yellow sides, and one blue side. If your spin lands on blue, or if the blue side of the cube comes up, you win a stuffed animal.
a. Draw both an area diagram and a tree diagram to represent the sample space for this game.
b. Which diagram was easier to make? From which diagram is most useful for showing what result is most likely?
c. What is the probability of winning a stuffed animal. P(blue)?
d. About how many times would you expect to have to play this game in order to win a stuffed animal?
e. If it cost a dollar to play the game and the stuffed animal would have been purchased at the Bob's Bargain basement for $3.50, was it worth the money? Explain.
f. Suppose that you know that Tyler won a stuffed animal. Talk with your team and use your area model to help you figure out what the probability that he started off with heads is? Be prepared to share your ideas with the class.
g. In part (f), you calculated the conditional probability of having started with heads given that you knew you rolled or spun blue. Which diagram shows this relationship most clearly? Explain.
10-28. Imagine rolling two standard six-sided dice. Use an area model to represent all possible combinations of numbers.
a. What is the probability of rolling a sum of eight?
b. Maribelle has a sum of eight. What is the probability that she rolled a three and a five?
10-29. BUILD-A-FARM
In the children's game, "Build-a-Farm," each player first spins a spinner. Half of the time the spinner comes up red. Half of the time the spinner comes up blue. If the spinner is red, you reach into the red box. If the spinner is blue, you reach into the blue box. The red box has 10 chicken counters, 10 pig counters, and 10 cow counters, while the blue box has 5 chicken counters, 4 pig counters, and 1 cow counter.
a. Sketch an area diagram for the situation where a child spins and then draws. Note that the parts corresponding to the two boxes of animal markers will be quite different.
b. Shade the parts of the diagram corresponding to getting a pig counter. What is P(pig)?
c. Find P(cow counter), the probability of getting a cow counter.
d. Now, supposing you got a cow counter, what is the probability that your spin was red?
e. Find the conditional probability that if you got a cow counter, your spin was blue.
10-30. If Letitia studies for her math test tonight, she has an 80% chance of getting an A. If she does not study, she only has a 10% chance. Whether she can study or not depends on whether she has to work at her parents' store. Earlier in the day, her father said he was feeling sick and there is a 50% chance he will not be able to work tonight.
a. Draw a diagram for the situation.
b. Find the probability that Letitia gets an A.
c. What are the chances that Letitia studied, given that she got an A?
10-31. La Troy has been studying very hard for his English test. He thinks that given any question, he has a 99% chance of getting it right.
a. What is the probability that he gets the first three questions right?
b. If the test has fifty questions, what is the probability that he gets them all correct?
c. Suppose La Troy wanted a 90% chance he would get every question on the test correct. What would his chances of getting each question correct have to be?
d. If you did not do so already, write an equation to represent part (c) and solve it.
10-32. When he was in the first grade, Harvey played games with spinners. One game he especially liked had two spinners and several markers that you moved around a board. You were only allowed to move if your color came up on both spinners. Harvey always chose purple because that was his favorite color.
a. What was the probability that Harvey could move his marker?
b. What was the probability for the better choice of color?
c. How often did no one get to move?
d. There are at least two ways to figure out part (c). Discuss your solution method with your team and show a second way to solve part (c).
[One spinner is 1/4 Green, 1/4 Purple, 1/2 Yellow. The other spinner is 1/3 Green, 1/3 Purple, and 1/3 Yellow.]
10-33. Sometimes it is easier to figure out the probability that something will not happen than the probability that it will. Show two ways to solve the problem below and decide which way you prefer and explain why.
Crystal is spinning the spinner at right and claims she has a good chance of having the spinner land on red at least once in three tries. What is the probability that the spinner will land on red at least once in three tries?
[Spinner is 1/3 Red, 1/3 Blue, and 1/3 Yellow]
10-34. Eddie is arguing with Tana about the probability of flipping three coins. They decided to flip a penny, nickel, and a dime. If they flipped three coins, would a tree diagram or an area model be better for determining the sample space? Justify your answer.
a. How many outcomes are in the sample space?
b. What is the probability of getting each of the following outcomes? Show all of your reasoning.
i. Three Tails? ii. At least two tails?
iii. Exactly two tails? iv. At least one head?
c. How would the probabilities change if Tana found out that Eddie was using weighted coins, coins that were not fair, so that the probability of getting heads for each coin was 4/5 instead of 1/2? Would this change the size of the sample space? Recalculate the probabilities in part (b) based on the new information.
10-35. A spinner has just two colors, red and blue. The probability the spinner will land on blue is x.
a. What is the probability it will land on red?
b. Sketch an area diagram for spinning twice.
c. When it is spun twice, what is the probability it will land on the same color both times?
d. Given that it lands on the same color twice, what is the probability that it landed on blue both times?
10-36. On another spinner, blue occurs a fraction x of the time, while the red and green portions have equal area. There are no other colors.
a. Represent the probability that the spinner will land on green.
b. Sketch an area diagram for spinning twice.
c. Shade the region on your area diagram corresponding to getting the same color twice.
d. What is the probability that both spins give the same color?
e. If you know that you got the same color twice, what is the conditional probability that the color was blue?
10-37. LEARNING LOG
Looking back at the problems in the last two lessons, examine the use of systematic lists, tree diagrams, and area models. For what types of problems or subproblems is a systematic list most helpful? Tree diagrams? Area models? Reflect on these strategies. Explain how they can be used to represent the sample space in a problem and how they can help find probabilities. Title this entry "Representing Sample Spaces using Systematic Lists, Tree Diagrams, and Area Models" and label it with today's date.
10-38. Guess what? Another spinner. This time the spinner lands on red half of the time and on green one third of the time. The rest of the time it lands on blue.
a. Draw an area diagram for spinning twice and shade the region that corresponds to getting the same color on both spins, for each of the three colors.
b. Suppose you know that the spinner landed on the same color twice. What is the probability that that color was green?
10-39. What is the probability that x² + kx + 12 will factor if 0 ≤ k ≤ 8 and k is an integer? Make an organized list (sample space) to help you determine the probability.
10-40. You decide to park your car in a parking garage that charges $3.00 for the first hour and $1.00 for each hour (or any part of an hour) after that.
a. How much will it cost to park your car for 90 minutes?
b. How much will it cost to park your car for 118 minutes? 119 minutes?
c. How much will it cost to park your car for 120 minutes? 121 minutes?
d. Graph the cost in relation to the length of time your car is parked.
e. Is this function continuous?
10-41. Victoria is playing with her balance scale and balances 3 blue blocks and 2 red blocks on one side with 5 blue blocks and a red block on the other. Later, her brother balances 2 red blocks and a blue block with a 40-gram weight. How much does each block weigh?
10-42. Factor and reduce to simplify:
3x+6 . Justify each step. [` symbol is being used to let the equation be formatted properly]
`````````````````````````````x²+7x+10
10-43. Simplify and add
x-4 +
4x+12 . Justify each step. [` symbol is being used to let the equation be formatted properly]
```````````````````x+2 `x²+5x+6
10-44. After paying $20,000 for a car you read that this model has decreased in value 15% per year over the last several years. If this trend continues, in how many years will your car be worth only half of what you paid for it?
10-45. Change the angle measurements below from radians to degrees or from degrees to radians. [π = pi symbol] [` symbol is being used to let the equation be formatted properly]
a. 144°
b. 300°
c.
5π``9
d.
17π``12
e.
19π ``2
f. 220°
10-46. Antonio and Giancarlo are playing a board game that uses three spinners, which are shown in the diagrams at right. The first is two-thirds red and one third green. The second is all red and split into four equal parts with the numbers 0, 2, 4, and 6 in each quarter. The third is all green and split into three unequal sections, marked with a 6, a 4, and a 2. On each turn the player has to spin two of the spinners to determine how far he gets to move his marker. The color that comes up on the first spinner determines which of the other two spinners the player has to spin to get the number of spaces to move.
a. Make an area model or tree diagram to represent the possible outcomes.
b. What is the probability that Giancarlo will get to move his marker 4 or 6 spaces?
c. What is the probability he will have to stay put?
d. What is the probability that he will get to move?
e. Antonio was moving his marker 2 spaces. What is the probability that he had spun green on the first spinner?
[Spinner #3: Piece 6 is 1/4, Piece 4 is 2/5, Piece 2 doesn't specify <Although it looks 1/3 in size>
10-47. Rearrange each equation into a more useful form for graphing. State the locator point and draw the graph.
a. y=2x²+8x+5 b. x²+y²-2x-6y=0
10-48. Solve the system of equations at right. [Typed below due to formatting on forum]
8x-3y-2z=-8
-2x+8y+7z=26
4x+y-5z=23
10-49. Divide: (x³-2x²+25x-50)/(x-2). [³=cubed ²=squared]
10-50. Use your answer from the previous problem to solve x³-2x²+25x-50=0. [³=cubed ²=squared]
10-51. Write the reciprocal of each of the following in simplest form. [ √ is the squareroot symbol]
a. √2+√2i b. 1-√3
10-52. If cosØ=
8 , find sinØ.
````````````17
This post was edited by WGUS on Mar 17 2015 12:33am