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Statistical Probabilities and Distributions

Odd Solutions for Stat. Prob./Dist. Homework 2

  1. Imelda Marcos has 2003 pairs of shoes. In how many different ways can she select a left shoe then a right shoe?

    Answer: 2003 × 2003 = 4012009.

  2. Scanti Lee Clad has 15 short shorts and 11 sleeveless blouses. In how many different ways could she select a shorts-blouse combination.

  3. Lisa visits Pet Refuge and finds 21 dogs and 13 cats she likes. In how many ways could she select a dog or a cat? In how many ways could she select a dog and a cat?

    Answer: or: 21+13 = 34;   and: 21 × 13 = 273.

  4. The MSC summer reading list contains 12 science books and 15 math books.
    1. In how many different ways could a student select a science or math book?
    2. In how many different ways could a student select a science and then a math book?
    3. In how many different ways could a student select a math and then another math book?

  5. Fix Or Repair Daily manufactures light trucks with three different body styles, five different colors of paint, and six different interior colors. Compare the number of trucks necessary to exhibit an example of each with the number of possible varieties.

    Answer: combos: 3 × 5 × 6 = 90;   examples: max(3, 5, 6) = 6.

  6. Because of a two hour fog delay only 8 sophomores showed up for math class. Thus the students sit in two table groups of four students each. (Assume each chair gets numbered from 1 to 8.)
    1. In how many different ways could a student be selected to occupy chair 1?
    2. After chair 1 is occupied, how many different ways are there of seating someone in chair 2?
    3. In how many different ways could chair 1 and chair 2 be filled?
    4. If chairs 1 and 2 are occupied, how many ways could chair 3 be filled?
    5. In how many different ways could chairs 1, 2, and 3 be filled?
    6. In how many different ways could all eight chairs be filled?

  7. Telephone numbers in the United States and Canada have three groups of digits which meet certain requirements:
    1. Area Code: 3 digits, the first of which is neither 0 nor 1.
    2. Exchange: 3 digits, the first of which is neither 0 nor 1.
    3. Line Number: 4 digits, with 0000 disallowed.

    1. How many possible area codes are there?
    2. How many possible exchanges are there?
    3. How many possible line numbers are there?
    4. How many valid 10-digit phone numbers are there?
    5. What is the probability that a random 10-digit number is a valid phone number?

    Answer: a) 8×10×10=800;
    b) 8×10×10=800;
    c) 104-1=9999 (not 9000!)
    d) 800×800×9999 = 6,399,360,000
    e) 63.99%

  8. What is the probability that a card drawn at random from a shuffled deck of 52 normal playing cards is a Heart or a Face card? Be sure to avoid double counting!

  9. In how many ways could you arrange the following?
    1. Four notebook sections from a set of six sections?
    2. Ten homeworks from a set of twelve homeworks?
    3. Five tests from a set of eight tests?
    4. All 20 questions from a set of 20 questions?

    Answer: a) 6P4 = 360.
    b) 12P10 = 12!/2=239,500,800.
    c) 8P5 = 6720.
    d) 20!=2.432 902 008 ×1018.

  10. How many arrangements can be made from the 26 letters in the English alphabet by using:
    1. 2 different letters?
    2. 3 different letters?
    3. 4 letters without replacement?
    4. 4 letters with replacement?

  11. Fourteen people try out for a baseball team. In how many different ways could they select:
    1. the pitcher and the catcher?
    2. the three outfielders, after the pitcher and catcher have been selected?
    3. the four infielders (1st, 2nd, SS, and 3rd), after the other five team members have been selected?

    Answer: a) 14 × 13 = 182.
    b) 12P3 = 1320.
    c) 9P4 = 3024.

  12. Teacher Thelma says "You may work these five problems in any order you choose." There are 30 students in the class. It is possible for all 30 students to work the problems in a different order? Justify your answer. Don't just answer yes or no.

  13. A 6-letter permutation is selected at random from the letters UNITED. What is the probability that:
    1. The third letter is I and the last letter is T?
    2. The second letter is a vowel and the third is a consonant?
    3. The second and third letters are both vowels?
    4. The second letter is a consonant and the last letter is E?
    5. The second letter is a consonant and the last letter is T?

    Answer: a) 1/6 × 1/5 = 1/30.
    b) 3/6 × 3/5 = 9/30 = 3/10 = 0.3.
    c) 3/6 × 2/5 = 1/5 = 0.2.
    d) 3/6 × 1/5 = 3/30 = 0.1.
    e) 2/6 × 1/5 = 1/15.

  14. Six girls start playing a volleyball game.
    1. In how many ways could the six positions be filled?
    2. In how many ways could the six positions be filled, if Nikki must be server?
    3. If the positions are selected at random, what is the probability that Nikki will be server?
    4. Express the probability in part c above as a percentage.

  15. Twelve sophomores line up for a fire drill.
    1. How many possible arrangements are there?
    2. How many arrangements have David and Steph next to each other?
    3. If they line up at random, what is the probability that David and Steph will be next to each other?

    Answer: a) 12! = 479,001,600.
    b) 2 × 11! = 79,833,600.
    c) 2 × 11!/12! = 2/12 = 1/6.

  16. How many different permutations are there in the letters of: BUBBLES?

  17. How many different permutations are there in the letters of: DENNIS?

    Answer: 6!/2! = 360.

  18. How many different circular permutations can be made from: ARITHMETIC.

  19. How many different ways can five boys and five girls sit alternately around a merry-go-round?

    Answer: 5! × 4! = 120 × 24 = 2880.

  20. Four boys and four girls hold hands in a circle, with boys and girls alternating. In how many different ways can they be arranged, if it doesn't matter which side someone is on?

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