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

  • 0.5000
  • 0.9329
  • 0.0378
  • 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=x) 0.03 0.50 0.24 ? 0.07 0.04
  • Describe the random variable X in words.
  • Find the probability that a customer rents three DVDs.
  • Find the probability that a customer rents at least 4 DVDs.
  • Find the probability that a customer rents at most 2 DVDs.

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=x) 0.35 0.25 0.20 0.10 0.05 0.05
  • At which store is the expected number of DVDs rented per customer higher?
  • 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.
  • 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.
  • Which of the two video stores experiences more variation in the number of DVD rentals per customer? How do you know that?

Partial Answer:
A: X = the number of DVDs a Video to Go customer rents
B: 0.12
C: 0.11
D: 0.77

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.
  • Find the expected value for this game (expected net gain or loss).
  • Explain what your calculations indicate about your long-term average profits and losses on this game.
  • 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 fair six-sided 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)
  • Write the description of the random variable X. What are the values that X can take on? Find the values of p and q.
  • Find the probability that the first occurrence of event F (rolling a “4” or “5”) is on the second trial.
  • How many trials would you expect until you roll a “4” or “5”?

A: X can take on the values 1, 2, 3, .... p = 2/6, q = 4/6
B: 0.2222
C: 3

**Exercises 38 - 43 contributed by Roberta Bloom

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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Source:  OpenStax, Collaborative statistics (custom lecture version modified by t. short). OpenStax CNX. Jul 15, 2013 Download for free at http://cnx.org/content/col11543/1.1
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