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4.1 Homework (modified r. bloom)  (Page 6/6)

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Suppose we randomly survey 20 pages. We are interested in the number of pages that advertise footwear. This time, each page may be picked more than once.

  • D

    How many pages do you expect to advertise footwear on them?

  • E

    Is it probable that all 20 will advertise footwear on them? Why or why not?

  • F

    What is the probability that less than 10 will advertise footwear on them?

  • G

    Suppose that a page may be picked more than once. We are interested in the number of pages that we must randomly survey until we find one that has footwear advertised on it. Define the random variable X and give its distribution.

  • H

    Do you expect to survey more than 10 pages in order to find one that advertises footwear on it? Why?

  • I

    What is the probability that you only need to survey at most 3 pages in order to find one that advertises footwear on it?

  • J

    How many pages do you expect to need to survey in order to find one that advertises footwear?

  • D

    3.02

  • E

    No

  • F

    0.9997

  • H

    0.2291

  • I

    0.3881

  • J

    6.6207 pages

Suppose that you roll a fair die until each face has appeared at least once. It does not matter in what order the numbers appear. Find the expected number of rolls you must make until each face has appeared at least once.

Try these multiple choice problems.

For the next three problems : The probability that the San Jose Sharks will win any given game is 0.3694 based on their 13 year win history of 382 wins out of 1034 games played (as of a certain date). Their 2005 schedule for November contains 12 games. Let X size 12{X} {} = number of games won in November 2005

The expected number of wins for the month of November 2005 is:

  • A

    1.67

  • B

    12

  • C

    382 1043

  • D

    4.43

D: 4.43

What is the probability that the San Jose Sharks win 6 games in November?

  • A

    0.1476

  • B

    0.2336

  • C

    0.7664

  • D

    0.8903

A: 0.1476

Find the probability that the San Jose Sharks win at least 5 games in November.

  • A

    0.3694

  • B

    0.5266

  • C

    0.4734

  • D

    0.2305

C: 0.4734

For the next three questions : The average number of times per week that Mrs. Plum’s cats wake her up at night because they want to play is 10. We are interested in the number of times her cats wake her up each week.

In words, the random variable X size 12{X} {} =

  • A

    The number of times Mrs. Plum’s cats wake her up each week

  • B

    The number of times Mrs. Plum’s cats wake her up each hour

  • C

    The number of times Mrs. Plum’s cats wake her up each night

  • D

    The number of times Mrs. Plum’s cats wake her up

A: The number of times Mrs. Plum's cats wake her up each week

Find the probability that her cats will wake me up no more than 5 times next week.

  • A

    0.5000

  • B

    0.9329

  • C

    0.0378

  • D

    0.0671

D: 0.0671

People visiting video rental stores often rent more than one DVD at a time. The probability distribution for DVD rentals per customer at Video To Go is given below. There is 5 video limit per customer at this store, so nobody ever rents more than 5 DVDs.

X 0 1 2 3 4 5
P(X) 0.03 0.50 0.24 ? 0.07 0.04

  • A

    Describe the random variable X in words.

  • B

    Find the probability that a customer rents three DVDs.

  • C

    Find the probability that a customer rents at least 4 DVDs. Write your answer using proper notation.

  • D

    Find the probability that a customer rents at most 2 DVDs. Write your answer using proper notation.

Another shop, Entertainment Headquarters, rents DVDs and videogames. The probability distribution for DVD rentals per customer at this shop is given below. They also have a 5 DVD limit per customer.

X) 0 1 2 3 4 5
P(X) 0.35 0.25 0.20 0.10 0.05 0.05

  • E

    At which store is the expected number of DVDs rented per customer higher?

  • F

    If Video to Go estimates that they will have 300 customers next week, how many DVDs do they expect to rent next week? Answer in sentence form.

  • G

    If Video to Go expects 300 customers next week and Entertainment HQ projects that they will have 420 customers, for which store is the expected number of DVD rentals for next week higher? Explain.

  • H

    Which of the two video stores experiences more variation in the number of DVD rentals per customer? How do you know that?

Solution will be posted on the instructor's website for this class.

A game involves selecting a card from a deck of cards and tossing a coin. The deck has 52 cards and 12 cards are "face cards" (Jack, Queen, or King)The coin is a fair coin and is equally likely to land on Heads or Tails

  • If the card is a face card and the coin lands on Heads, you win $6
  • If the card is a face card and the coin lands on Tails, you win $2
  • If the card is not a face card, you lose $2, no matter what the coin shows.

  • A

    Find the expected value for this game (expected net gain or loss).

  • B

    Explain what your calculations indicate about your long-term average profits and losses on this game.

  • C

    Should you play this game to win money?

The variable of interest is X = net gain or loss, in dollars

The face cards J, Q, K (Jack, Queen, King). There are(3)(4) = 12 face cards and 52 – 12 = 40 cards that are not face cards.

We first need to construct the probability distribution for X. We use the card and coin events to determine the probability for each outcome, but we use the monetary value of X to determine the expected value.

Card Event $X net gain or loss P(X)
Face Card and Heads 6 (12/52)(1/2) = 6/52
Face Card and Tails 2 (12/52)(1/2) = 6/52
(Not Face Card) and (H or T) –2 (40/52)(1) = 40/52
  • Expected value = (6)(6/52) + (2)(6/52) + (–2) (40/52) = –32/52
  • Expected value = –$0.62, rounded to the nearest cent
  • If you play this game repeatedly, over a long number of games, you would expect to lost 62 cents per game, on average.
  • You should not play this game to win money because the expected value indicates an expected average loss.

You buy a lottery ticket to a lottery that costs $10 per ticket. There are only 100 tickets available be sold in this lottery. In this lottery there is one $500 prize, 2 $100 prizes and 4 $25 prizes. Find your expected gain or loss.

Start by writing the probability distribution. X is net gain or loss = prize (if any) less $10 cost of ticket

X = $ net gain or loss P(X)
$500–$10=$490 1/100
$100–$10=$90 2/100
$25–$10=$15 4/100
$0–$10=$–10 93/100)

Expected Value = (490)(1/100) + (90)(2/100) + (15)(4/100) + (–10) (93/100) = –$2. There is an expected loss of $2 per ticket, on average.

A student takes a 10 question true-false quiz, but did not study and randomly guesses each answer. Find the probability that the student passes the quiz with a grade of at least 70% of the questions correct.

  • X = number of questions answered correctly
  • X~B(10, 0.5)
  • We are interested in AT LEAST 70% of 10 questions correct. 70% of 10 is 7. We want to find the probability that X is greater than or equal to 7. The event "at least 7" is the complement of "less than or equal to 6".
  • Using your calculator's distribution menu: 1 – binomcdf(10, .5, 6) gives 0.171875
  • The probability of getting at least 70% of the 10 questions correct when randomly guessing is approximately 0.172

A student takes a 32 question multiple choice exam, but did not study and randomly guesses each answer. Each question has 3 possible choices for the answer. Find the probability that the student guesses more than 75% of the questions correctly.

  • X = number of questions answered correctly
  • X~B(32, 1/3)
  • We are interested in MORE THAN 75% of 32 questions correct. 75% of 32 is 24. We want to find P(X>24). The event "more than 24" is the complement of "less than or equal to 24".
  • Using your calculator's distribution menu: 1 - binomcdf(32, 1/3, 24)
  • P(X>24) = 0.00000026761
  • The probability of getting more than 75% of the 32 questions correct when randomly guessing is very small and practically zero.

Suppose that you are perfoming the probability experiment of rolling one die. Let F be the event of rolling a "4" or a "5". You are interested in how many times you need to roll the die in order to obtain the first “4 or 5” as the outcome.

  • p = probability of success (event F occurs)
  • q = probability of failure (event F does not occur)

  • A

    Write the description of the random variable X. What are the values that X can take on? Find the values of p and q. What is the appropriate probability distribution for X?

  • B

    Find the probability that the first occurrence of event F (“4” or “5”) is on the first or second trial.

  • C

    Find the probability that more than 4 trials are needed to obtain the first “4” or “5” when rolling the die.

Solution will be posted on the instructor's website for this class.
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Source:  OpenStax, Collaborative statistics homework book: custom version modified by r. bloom. OpenStax CNX. Dec 23, 2009 Download for free at http://legacy.cnx.org/content/col10619/1.2
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