Understanding Data Analysis and Counting Principles for Students
School
Yeshivas Ohr Yissochor**We aren't endorsed by this school
Course
BUSINESS 101
Subject
Computer Science
Date
Dec 12, 2024
Pages
2
Uploaded by mqfyqyp
Data Analysis 13. Nationally, what would be the percentage of 12- to 17-years-olds that had tried marijuana in the past month for 2011? 5.6% 6.0% 6.5% 6.6% 6.8% BEO®®E® Quantity A Quantity B 14. 2011 marijuana users in West 2011 marijuana users in South COUNTING There are many problems that ask how many ways something can happen. Although it's not as simple as what you learned in kindergarten, we call that counting, The most fundamental problems involve combining situations. Basic Counting Principle If there are m ways to do Xand » ways to do Y] then there are m X » ways to do Xand Y. ( | I | | | f \ Example 1 A combination door lock has a two symbol access code. The first digit is A, B, C, or D. The second is 1, 2, 3, 4, or 5. How many different access codes are there? How many access codes have only consonants and even numbers in them? @ 20 16 © 12 @ 9 ® 6 Solution: According to the counting principle, since there are four choices for the first sym- bol (A, B, C, or D) and five choices for the second symbol (1, 2, 3, 4, 5), there are 4 X 5 = 20 possible access codes. We list them all in hopes you see why this principle works. 2nd Symbol 12345 _|A| A1 | A2 |A3|A4 A5 é B|B1|B2|B3|B4|B5 ‘(’E clct|cz2|c3|ca|cs "|p|D1|D2|D3|D4|D5 157
158 GRE Math Workbook e Watch out for double counting. It is easy to overestimate the result by counting some values twice (or more). If the first symbol has to be a consonant, there are only three choices (B, C, or D). The second symbol has to be even so there are two choices (2 or 4). Hence, there are 3 X 2 = 6 possible access codes that are made of a consonant and an even number. Choose E. Gary has 4 pairs of pants, 4 ties, and 4 shirts. Quantity A Quantity B Number of outfits if he Number of outfits if loses a tie and a shirt he loses 2 pants Solution: The counting principle applies even to more than two situations. Here Gary has 4 X 4 X 4 outfits to begin with, since he has a choice of 4 pants, 4 ties, and 4 shirts. In both cases, he loses two articles of clothing so the result is the same, right? Wrong. If he loses a tie and a shirt, he now has 4 X 3 X 3 = 36 outfits. If he loses two pants, he now has 2 X 4 X 4 = 32 outfits. Choose A. Example 3 If xand y are integers so that 1 = x < 5 and 2 < y < 7, how many possible values could xy have? @ 30 ® 25 © 20 @ 16 ® 13 Solution: Since1 < x < 5 and x is an integer, there are four choices for x: 1, 2, 3, and 4. Since 2 < y < 7 and yis an integer, there are 4 choices for y: 3, 4, 5, and 6. So by the count- ing principle, we expect there are 4 X 4 = 16 possible values for xy. Unfortunately, this is wrong. There are 16 possible products, but some of them have the same value. Let’s see: 1X3=3 2X3=6 3X3=9 4 xX3=12 1X4=4 2X4=8 3X4=12 4x4=16 1X5=5 2x5=10 3xX5=15 4 X5=20 1X6=6 2X6=12 3xXx6=18 4 X6=24 As you can see from the table, some values are repeated. So we actually have only 13 dif- ferent possible values for xy. You don't actually have to check all these possible values. After determining there were at most 16 possible values and knowing that some of them were repeated, you would know the actual answer was less than 16. Thus the only possible answer is E. Venn Diagrams There are 100 students at Baltimore High School—60 ride the bus and 60 walk. How is that possible? Wouldn't that make 120 students? No, because 20 do both. They walk in the morning but take the bus after school.