Illinois State University Mathematics Department

 MAT 312: Probability and Statistics for Middle School Teachers Spring 1999 9:35 - 10:50 am TR STV 350A Dr. Roger Day (day@math.ilstu.edu)

 Possible Solutions to Quiz #3 look at Quiz #3 look at course grades

 1,2,3,4: 1 point each 5: 2 points 6: 1 point each 7: 2 points Full credit will be awarded for correct responses. It is to your advantage to show steps leading to your solution and to include brief explanations where appropriate. Question 7 requires an explanation as well as a numerical response.

approximately 2% to 3%

Use the following sets for questions 1 through 4:
 Set I: {a,c,g,h,i,l,o,r,y} Set II: {e,i,n,o,q,s,t,u} Set III: {a,d,e,m,o,w}

1. A letter is to be chosen from Set I or from Set II. How many choices are there?

9+8-2=15

2. Two-letter words (meaningful or otherwise) are to be created using a letter from Set I as the first letter of the word and a letter from Set II as the second letter of the word. How many different two-letter words can be created in this manner?

9*8=72

3. Three-letter sets are to be created using the letters from Set II with no repetition allowed. For example, {e,i,n} can be created, but not {e,e,e} nor {e,e,i}. Note, also, that the set {e,i,n} is equivalent to the set {n,e,i}. How many unique three-letter sets can be created?

C(8,3)=56

4. Suppose we carried out the following procedure:

• Step A: Select four different letters from Set III.
• Step B: Arrange the four letters in as many ways as possible.

If we carried out this procedure as many times as possible, how many unique 4-letter arrangements could be made using the letters from Set III?

P(6,4)=360

5. The prom supervisor at a local high school asked for volunteers for next year's prom committee. There were seven 10th-grade and five 9th&endash;grade volunteers. The advisor only needed three from each of the two classes. How many 6-person prom committees could be formed under these conditions?

C(7,3)*C(5,3)

6. The design of a quilt for a newborn baby is shown here, composed of three parts: an inset, a body, and a border. Suppose each of the three parts of the quilt is to be a solid color, with each part a different color.

a. If there are seven colors to choose from, how many different quilts could be made?
7*6*5=210

b. The Morton Sewing Circle wants to sew at least 500 quilts like this, with no two alike. What is the minimum number of colors required that seamstresses must be able to choose from?

We need n so that n*(n-1)*(n-2) is greater than or equal to 500. The value n = 9 is the smallest positive integer that satisfies this requirement. Therefore, we need 9 colors.

7. Junior Samples was left in charge of the hat-check room at a recent square dance. Against one wall in the hat-check room was a rectangular array of numbered cubicles into which hats could be placed. A big sign hung over the cubicles. Here's what it said:

 Hat-Check Policy: No more than three hats in each cubicle!

During the night, Junior kept a tally sheet showing the time on the clock and the number of hats that were in check at that time. A portion of the tally sheet is shown below.

After the dance, Wally Woundhouse, Junior's supervisor, sauntered over to the hat-check room and surveyed Junior's tally sheet. He looked at the 9:05 entry and said, "Junior, if your figures are accurate, at precisely 9:05 I know that you violated the hat-check policy! You may have violated it some other time not shown here, but I can assure you that you did at 9:05!"

Based on Wally's statement, how many cubicles were there in the hat-check room? Explain how you know.

There are 50 cubicles in the hat-check room. The 9:05 entry that Wally read was the one on the sheet indicating a policy violation. If we subtract 1 from 151 we get 150, and 150 divided by 3 is 50. Any number of hats up to any including 150 was okay, but more than 150 violated the policy. We divide by 3 because there can be up to 3 hats per cubicle. With all 50 cubicles filled, there would be 150 hats.

Bonus!

Return to questions (1) through (4) and use the letters in each of Sets I, II, and III to spell a legitimate word. Each word must use all and only the letters from one set, with no repetition. One point will be awarded for one correct word from one set, Two points will be awarded for one correct word from each of two sets, and four points will be awarded for one correct word from each of the three sets.