What happens when you don t put the card back in the deck?

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1 Problem Solving: When One Thing Depends on Another Thing Problem Solving: When One Thing Depends on Another Thing What happens when you don t put the card back in the deck? We worked with coins, dice, and cards. We asked what our chances were for drawing an ace out of a deck of cards that was shuffled. Remember, the cards have to be mixed up, or random, for each card to have an equal chance of being drawn from the deck. We know there are four aces in a deck of 52 cards, which means that the probability is out of 52. Outcomes with drawing an ace: Total possibilities: 52 The probabilitiy is 52 = 0.077, or about 8%. Now let s think about the chances of pulling two aces in a row out of a deck and putting them face up on a table. We need to think about this problem carefully because we are not going to put the card from the first draw back in the deck. Example 1 shows how we solve this problem. We already know the chances of drawing an ace out of the deck on the first draw are 52, or Unit 7 Lesson 9 523

2 Example 1 shows that we would only have 51 cards left in the deck when we try to draw the second ace. In this case, drawing the second ace depends on drawing the first ace. That means that the probability of drawing the second ace depends on the card already drawn from the deck. The probability changes slightly because we have one less card for the second draw. We multiply the two fractions together to show what the probability of drawing two aces in a row would be. Example 1 Show the probability of drawing two aces in a row if the card from the first draw is not put back in the deck. Chances of selecting an ace on the first draw: Number of aces: Total possibilities: 52 Chances of selecting another ace on the second draw: Number of aces: 3 Total possibilities: 51 Probability of drawing two aces in a row: = 12 2,652, or The answer is about 0.5%, which is less than 1%. This is a small number. Pulling colored marbles from a jar is another way to think about this kind of probability. 52 Unit 7 Lesson 9

3 Suppose we had a jar with 10 marbles 6 green and yellow. What are the probabilities of first pulling out a green marble and then pulling out a yellow marble? The order is important. We need to pull out a green marble first, and then a yellow one. Pulling out the yellow marble depends upon what we do first. Example 2 shows how we calculate the probabilities. Again, the total number of marbles changes based on the first draw. That means on the second draw there will still be yellow marbles, but only 9 marbles altogether. Example 2 Show the probability of pulling out a green marble and then a yellow marble. Chances of drawing a green marble on the first draw: Number of green marbles: 6 Total possibilities: 10 Chances of drawing a yellow marble on the second draw: Number of yellow marbles: Total possibilities: 9 Probability of drawing a green marble and then a yellow marble in that order: = 2 90, or Probability of drawing a green marble. Probability of drawing a yellow marble. The decimal number is about 27%. The probability is about 27%. Unit 7 Lesson 9 525

4 What happens if we use just a few cards from the deck? We have been able to establish probabilities based on a full deck of cards. A regular deck of cards has 52 total cards and a unique mix of characteristics, numbers, colors, suits, etc. What happens if we use just a few cards from the deck? We would have to come up with a new set of probabilities based on the characteristics of the new set of cards. Suppose we have a deck of just 5 cards. The cards are the 2 of hearts, 2 of diamonds, 3 of clubs, of spades, and 5 of clubs. What is the probability of drawing two red cards if we replace the card drawn each time? What is the probability if we do not replace the first card drawn? To answer these questions, we have to think about probabilities in a different way. We have two red cards and three black cards, for a total of five cards. The first time we draw, we have a 2 5 chance of getting a red card. If we replace that card, we have a , or 25, or 16% chance of selecting two red cards in a row. We have created a new deck of cards with new probabilities. If we do not replace the first card, we have a 2 5 1, or 2 20, or 10% chance of selecting two red cards. This is different from the probabilities we computed for a regular deck of cards. The probabilities for that deck are 50% for selecting two red cards if we replace the first card drawn, and 2.5% for selecting two red cards if we do not replace the first card drawn. Problem-Solving Activity Turn to Interactive Text, page 273. Reinforce Understanding Use the mbook Study Guide to review lesson concepts. 526 Unit 7 Lesson 9

5 Homework Activity 1 Rewrite the numbers using scientific notation. Remember to round , , ,000 Activity 2 Each of the numbers has an error. They do not follow the rules for scientific notation. Write the letter on your paper (a and/or b) that tells the reason why it is not correct. Model (a) The decimal number is not between 1 and 10 (b) The base of the power is not 10 Answer: a. The number is not between 1 and Copyright 2010 by Cambium Learning Sopris West. All rights reserved. Permission is granted to reproduce this page for student use. Unit 7 Lesson 9 527

6 Homework Activity 3 A deck of cards is made up of the following: 3 kings king of hearts, king of diamonds, king of spades queens (all four suits) 2 jacks jack of hearts and jack of diamonds aces (all four suits) Tell the probability of each of the following. 1. What is the chance of selecting 2 queens from the deck if you replace the first card before drawing the second? 2. What is the chance of drawing a red card from the deck? 3. What is the chance of drawing a king of hearts?. What is the chance of drawing a king of hearts or a jack of diamonds from the deck? 5. What is the chance of drawing 2 jacks from the deck if you do not replace the first card before drawing the second? 6. What is the chance of drawing 2 red cards from the deck if you do not replace the first card before drawing the second? Activity Distributed Practice Solve Convert 0.3 to a fraction and a percent Convert 5 to a decimal number and a percent. 8. Convert 100% to a fraction and a decimal number Unit 7 Lesson 9 Copyright 2010 by Cambium Learning Sopris West. All rights reserved. Permission is granted to reproduce this page for student use.