out one marble and then a second marble without replacing the first. What is the probability that both marbles will be white?


 Richard Wells
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1 Example: Leah places four white marbles and two black marbles in a bag She plans to draw out one marble and then a second marble without replacing the first What is the probability that both marbles will be white? You can analyze the problem in this way For the first drawing, out of marbles are white The probability that the first outcome is white is, which reduces to Now assume that the first marble is white For the second drawing, out of the remaining marbles are white The probability that the second outcome is white is Multiply to find the probability of both events: Answer: The probability of Leah s drawing two white marbles is, or 0% Before first event After first event S K I L L P R A C T I C E Solve each problem Express all answers in fraction form Three coins are tossed at the same time What is the probability that all three coins will come up tails? Dan and Stan are playing a card game Of the seven cards Dan has in his hand, five are hearts If Stan takes two cards from Dan s hand without looking, what is the probability that both cards will be hearts? Use the spinner below to solve items to What is the chance of spinning a? Sherry spins the spinner twice What is the chance that she did not get a on either spin? Alan spins the spinner twice What is the chance that the sum of the spins is? Hasina has bills in her wallet: a $ bill, a $ bill, a $0 bill, and a $0 bill If she draws out two bills at random, what is the chance of her drawing out $0? Of the employees at Olympic Pharmacy, two will be chosen at random to work the night shift Ana hopes she will not be chosen The manager writes the employees names on cards and draws one out without looking The first name drawn is Grant What is the chance that Ana s name will be next? () () () () () Two sided dice are numbered from to If both dice are rolled and the two numbers added, there are possible sums Which of the following is a true statement? (Hint: Make a chart of all possible dice rolls) () The chance of getting a sum of is () The chance of rolling two s is () The chance of rolling two of the same number is () The chance of getting a sum of is () The chance of getting a sum of is Answers and explanations start on page PROGRAM : Statistics and Probability 9
2 Since the number represents a certain probability, the chance that an event will not happen is equal to minus the probability that an event will occur In this case,, the probability that the marble will not be red S K I L L P R A C T I C E Solve each problem Three coins are tossed in the air What is the chance that they will not all come up heads? The spinner below is divided into eight equal sections Ten items in a sale bin are priced: $ $ $ $ $ $ $ $ $ $ One item is chosen at random What is the probability that the item costs either $ or $? () () () () () 0 What is the probability of not spinning an odd number? T E C H N O L O G Y C o n n e c t i o n Many companies use contests to boost sales Usually, the consumer has to find a certain bottlecap or game piece inside the product to win a prize By law, companies must list the odds of winning a prize on the packaging or container Chunky Cookie Contest Rules and Odds RULES: Winning game pieces are randomly placed inside specially marked packages of Chunky Cookies If you find a bag of cookies with a specially marked game piece inside, scratch off the cookie chunks on the game piece to find out how much you ve won PROGRAM : Statistics and Probability Prize Number of Winning Pieces Estimated Odds Grand Prize $000 0 :,000 First Prize $00 00 :,000 Second Prize $00,000 :9,000 Third Prize Case of Cookies,000 :,000 Fourth Prize Bag of Cookies,0,000 : For example, if you buy a bag of Chunky Cookies, you have a in, or %, chance of winning fourth prize, another bag of cookies As you can see, more valuable prizes are more difficult to win If you bought 00 bags of cookies, about how many would have a game piece winning fourth prize? About how much harder is it to win $00 than to win $00? Answers and explanations start on page
3 PART TWO DIRECTIONS: Choose the one best answer to each of the following problems You may not use a calculator on these problems Carla is helping her daughter with a math problem: There are men with hats The hats are put into a big box What is the probability that, without looking, each man will pick out his own hat? () () () () () Not enough information is given With one roll of a sixsided die, what is the probability that Joel will not roll a three or a six? () in () in () in () in () in Question is based on the following drawing MATHEMATICS Avion Bird Club monitors bird migration One day 00 birds fly by the watch station: are ducks and are geese Which of the following expressions best represents how to calculate the probability that a passing bird is a goose? () () 00 () 00 () () 00 () () 00 Question is based on the following drawing The Altadena Food Bank sent jars of spaghetti sauce and 0 jars of tomatoes to a shelter What is the probability that the first jar the cook pulls out of the carton will be a jar of spaghetti sauce? () 0 () () () () 9 In a garment plant, % of the shirts sewn do not pass inspection and must be sold as seconds An inspector has 00 shirts to inspect What is the chance that a shirt chosen at random will not pass inspection? () in 00 () in 0 () in 0 () in () in 0 A spinner is divided into equal sections and numbered as shown What is the probability of the pointer stopping on a number less than? () 00% () % () 0% () % () % 0 Ian tells Ruth that he will buy her lunch if she draws one of the four aces on the first pick from a deck of cards What are the chances that Ian will have to buy Ruth lunch? () in () in () in () in () in
4 The students may be making progress Even though the test scores have declined, the margin of error for the test results is plus or minus % Depending on the ability of the test to accurately measure performance, students reading ability for 99 may have been as low as 0% and their reading ability for 999 may have been as high as % The teachers would be wise to give their observations at least as much weight as the test scores Yes Considering the margin of error, math performance for 99 could have been as high as 0%, and math performance for 999 may have been as low as % Even considering the possibility of error, the scores show an increase of at least percentage points Skill Practice, page 0 0, or % 0 Add the number of blue and green outcomes, and place the sum over 0, and 0 0, or 0% You can either count the spins that are not black, or count black and subtract from 0: 0, and 0, or 0% Your sections may be in a different order, but each color should take up about the same fraction of the total circle as shown to right Skill Practice, page () 0 Count the combinations that contain neither C nor F Then write a ratio comparing that number to 0, the number of possible combinations: 0 0, or 0% or % There are possible dice rolls Remember that the combination and is different than and, even though the sums are the same Count the number of rolls that total more than, and compare to, the number 0 of possible rolls:, or about % 00 or % Write a probability ratio and reduce: 00 00, or % Science Connection, page The possible pairings are R R, R,r, r R, and r r One out of the four offspring will be white Skill Practice, page 9 For each coin, the probability that the toss will come up tails is Multiply to find the combined probability: 0 Before a card is drawn, out of cards are hearts Assume the first card is a heart Of the cards that are left, out of are hearts Multiply the two probabilities: 0 0 Of the equal sections, are labeled, and reduces to On either spin, there is a chance of not spinning Multiply: 9 The only way to get a sum of in two spins is by spinning two s The question is really asking, What is the probability of spinning two s? On any spin, the probability 9 of spinning a is Multiply: Hasina needs to draw the $0 and $0 bills in order to end up with $0 The possibility of getting either the $0 or $0 bill on the first draw is, or Assume she gets either of those bills The chance of drawing out the other on the second draw is Multiply: () After Grant s name is drawn, cards are left, and only one is Anna s () The chance of getting a sum of is Out of the possible combinations, there are only two that total : and Two out of reduces to Skill Practice, page The chance of all coming up heads is Subtract from The difference is () A total of six cards are labeled with either a or a Write the probability ratio and reduce: 0 Four sections are labeled with even numbers, and four are labeled with odd The probability is the same for both Write the probability ratio and reduce: Technology Connection, page About of the 00 bags would have a fourthplace game piece: 00 It is about twice as hard (half as likely) to win $00 as to win $00, since :,000 is half the frequency of :9,000 GED Practice, pages () One dime among four coins has a in chance of being randomly chosen: () () Wearhouse Focus on the data in the Medium column, rounding the statistics: (0 0 0) 0 Since Wearhouse sold 0 mediumsized sweaters, it sold about twice as many as the other stores combined You can also use number sense The store that sold twice as many as all the others combined would have to be the store with the greatest sales of that size () in The chance of getting a,000 collector s card is 0,000, which reduces to () Note that you want the chance of Lee picking an orange that is not starting to rot Find the probability of fresh oranges () among a total of oranges: ANSWER KEY
5 MATHEMATICS () % Waldo has chances out of 00 for the winning ticket, so he has chances out of 00, or a % probability of holding the winning ticket () The ratio of blue tokens to white tokens is probably : By sampling the contents of the bag, Mike can predict its contents By analyzing the data from the experiment, Mike can see that there are probably more blue tokens than white tokens in the bag, eliminating choices () through () Since the ratio of blue to white tokens during the experiment is 9:, which reduces to :, Mike can conclude that the ratio of tokens in the bag is probably : () There are 0 tiles in the box, and 0 are red Thus, 0 0 of the tiles are red, which reduces to () There are tiles that are either red or white, so out of 0 tiles are red or white The chance of drawing a red or white tile is 0, which reduces to 9 () Add 0 red tiles For the chance of drawing red to be 0%, half of the tiles in the box must be red There are already white and blue, for a total of 0 nonred tiles Adding 0 red to the 0 red already in the box would increase the total tiles in the box to 0, and the total red tiles to 0 0 red tiles out of 0 would be 0% red 0 () 9 There are two possible chances to pick a red candy out of nine remaining candies Be sure to subtract the one chosen candy from the original number of candies in the jar: ( ) 9 () Each spin has the same chance of being an A, a oneinsix probability Since this is a dependent event (spinning one A after the other), the ratios are multiplied: () % To win a prize, Karen will have to spin B on her second spin, a probability of in Her chance of not spinning B is in, or, which equals % () 0 to 0 The number of voters who said yes to the proposition was 00 The survey s margin of error is % (plus or minus two percentage points of 00, which is 0 voters), making the range 00 minus 0 to 00 plus 0, or 0 to 0 () The probability that the first man will get the right hat is Now men and hats are left The probability that the second man who tries will get his own hat is The probability for the third man is Of course, if the first three men get their own hats, then the fourth or last man must get his hat, the only hat left The total probability is found by multiplying all these events together (), or in () 00 This setup solution can be read, There are out of 00 chances that the bird is a goose Notice that the answer choices were not reduced in this case () % Eight of the sections are labeled with numbers less than (,,, or ):, which is a % chance () in Rethink the problem: Four of the six faces are not or, so the probability is in Reduce to in () There are chances out of that a jar of spaghetti sauce will be chosen: (0) 9 () in 0 If % of the shirts do not pass inspection, then out of 00, or out of 0, do not pass 0 () in The chances of picking any one of the aces are in, or in Alternate Math Formats, pages $0 $ $0 ($9$$$$) $0 $0 Put the weights in order and find the middle value There are 000 cards in all The chance of drawing a Dodger card is 0 out of 000, or in There are red sections and white sections, so the number of favorable outcomes is The chance of drawing either of the colors is out of, or Add the temperatures and divide the total by, the number of measurements taken Any value between $0 and $00 would be correct Note that the points on the graph form a line Mentally connect the points and determine where the line would cross the hour mark If you gridded in a value between 0 and 00, consider your answer correct Any value between 0 and 0 would be correct Each year the enrollment increases by about 0% A good estimate for Year is an enrollment of players, an increase of 0% from Year Consider an answer between 0 and 0 correct 9 Any value between and 0 would be correct As the price per family ticket increases, the number of tickets sold decreases Mentally draw a line through the existing points If the ticket price is raised to $0, the number of tickets sold will probably be between and 0 PROGRAM : INTRODUCTION TO ALGEBRA Skill Practice, page
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