HW1

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Des Moines Area Community College *

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330

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Statistics

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Apr 3, 2024

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docx

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1. A coin is tossed three times, and the sequence of heads and tails is recorded. (a) Determine the sample space, Ω. Ω = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} (b) List the elements that make up the following events: A = exactly two tails A = {TTH, THT, HTT} B = at least two tails B = {TTH, THT, HTT, TTT} C = the last two tosses are heads C = {HHH, THH} (c) List the elements of the following events: Ā = {HHH, HHT, HTH, THH, TTT} A  B = {TTH, THT, HTT, TTT} A  B = {TTH, THT, HTT} A  C =  2. Suppose that after 10 years of service, 35% of computers have problems with motherboards (MB), 30% have problems with hard drive (HD), and 20% have problems with both MB and HD. (a) What is the probability that a 10-year old computer has a problem with MB or HD? (b) What is the probability that a 10-year old computer still has a fully functioning MB and HD? Venn Diagram HD 0.2 MB 0.3 0.35 P(HD  MB) = P(HD) + P(MB) – P(HD  MB) = 0.3 + 0.35 – 0.2 = 0.45 P( MB  HD ) = P ( HD MB ) = 1 – P (HD  MB) = 1 – 0.45 = 0.55 ( De Morgan)

3. Twelve athletes compete in an archery event at the Olympics. (a) How many ways are there to award the Gold, Silver, and Bronze medals to these athletes? 3 steps: Assign Gold medal : 12 ways Assign Silver medal: 11 ways Assign Bronze medal: 10 ways Total: 12 * 11 * 10 = 1320 ways (b) How many ways are there to award 3 medals if we do not care about the color of the medal? Choose group of three out of 12 athletes : 12C3 = 220 ways (c) If we know the three individuals who got a medal, how many ways are there to distribute the Gold, Silver, and Bronze to these three individuals? Permutate 3 athletes in group of three: 3P3 = 6 ways 4. The AccessPlus system at ISU has the following policy for creating a password: • Passwords must be exactly 8 characters in length. • Passwords must include at least one letter (a-z, A-Z) or supported special character (@, #, $ only). All letters are case-sensitive. • Passwords must include at least one number (0-9). • Passwords cannot contain spaces or unsupported special characters. According to this policy, how many possible AccessPlus passwords are available? Round to the nearest trillion. (Hint: Count up the number of 8-character passwords that could be made, and then subtract off the number that don’t meet the requirement above) Choice for 1 character: 26 (a-z) + 26 (A-Z) + 10 (0-9) + 3 (@, #, and $) = 65 Total of 8-character passwords include these supported characters that could be made: 65 8 passwords Passwords of 8-character include these supported characters that don’t meet requirements All numbers:10 8 passwords

No number: 55 8 passwords possible AccessPlus passwords are available: 65 8 - 10 8 - 55 8 = 234.91 * 10 12  235 trillion 5. Harry Potter’s closet contains 12 brooms. 7 brooms are Comet 260 s, 4 brooms are Nimbus 2000 s, and 1 broom is a Firebolt . Harry, Ron, George and Fred want to sneak out in the middle of the night for a game of Quidditch. They are afraid to turn on the light in case they get caught. Harry reaches into the closet and randomly pulls 4 brooms out at once without looking. Total ways to choose 4 brooms from 12 brooms (order does not matter): 12C4 = 495 ways (a) What is the probability that all 4 chosen brooms are Comet 260 s? Event A: all 4 chosen brooms are Comet 260 s Ways to choose 4 from 7 Comet brooms: 7C4 = 35 ways P(A) = 35/495 = 0.0707 (b) What is the probability that Harry pulls out 1 Comet 260 , 2 Nimbus 2000 s, and 1 Firebolt broom? Event B: Harry pulls out 1 Comet 260, 2 Nimbus 2000s, and 1 Firebolt broom Total ways of event B: 3 steps Choose 1 Comet from 7 Comet brooms: 7C1 = 7 ways Choose 2 Nimbus from 4 Nimbus brooms: 4C2 = 6 ways Choose 1 Firebolt from 1 Firebolt broom: 1C1 = 1 ways Total: 7*6*1 = 42 ways P(B) = 42/495 = 0.0848 (c) What is the probability that at least 1 of the 4 chosen brooms is a Comet 260 ? Event C: at least 1 of the 4 chosen brooms is a Comet 260 C : all 4 chosen brooms are not Comet Total ways of C : choose 4 brooms from 5 brooms ( 4 Nimbus and 1 Firebolt) = 5C4 = 5 ways P ( C ) = 5/495 =0.0101 P(C) = 1 - P( C ) = 1 - 0.0101 = 0.9899 6. Suppose you have two urns with poker chips in them. Urn I contains two red chips and four white chips. Urn II contains three red chips and one white chip. You randomly select

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