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AP®︎/College Statistics

Course: AP®︎/College Statistics > Unit 7

Lesson 1: Estimating probabilities using simulation
Math
AP®︎/College Statistics
Probability
Estimating probabilities using simulation
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Theoretical and experimental probability: Coin flips and die rolls

AP.STATS:
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UNC‑2.A.4 (EK)
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UNC‑2.A.6 (EK)
Google Classroom
Probability tells us how likely something is to happen in the long run. We can calculate probability by looking at the outcomes of an experiment or by reasoning about the possible outcomes.

Part 1: Flipping a coin

question a
A fair coin has 2 sides (heads and tails) that are equally likely to show when the coin is flipped.
What is the theoretical probability that a fair coin lands on heads?
P, left parenthesis, start text, h, e, a, d, s, end text, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

question b
Dave flipped a coin 20 times and got heads on 8 of the flips.
Based on Dave's results, what is the experimental probability of the coin landing on heads?
P, left parenthesis, start text, h, e, a, d, s, end text, right parenthesis, approximately equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

question c
Why aren't the theoretical and experimental results the same?
Choose 1 answer:

question d
Dave continues flipping his coin until he has 100 total flips, and the coin shows heads on 47 of those flips.
Based on these results, what is the experimental probability of the coin landing on heads?
P, left parenthesis, start text, h, e, a, d, s, end text, right parenthesis, approximately equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

question e
What do you notice about the experimental probability after Dave continued flipping the coin?
Choose 1 answer:

Part 2: Rolling a die

question a
A fair die has 6 faces numbered 1 through 6 that are each equally likely to show when the die is rolled.
What is the theoretical probability that a fair die shows a 1, question mark
P, left parenthesis, 1, right parenthesis, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

question b
Dave is going to roll a die 60 times and see how often a 1 shows.
According to the theoretical probability, how many rolls should Dave expect to show a 1, question mark
approximately equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
rolls

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  • female robot ada style avatar for user Zhuo.Jessica.PS122
    Posted 6 years ago. Direct link to Zhuo.Jessica.PS122's post “How does probability rela...”
    How does probability relate to everyday life?
    (55 votes)
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    • blobby green style avatar for user James Boshart
      Posted 3 years ago. Direct link to James Boshart's post “Quantum Mechanics (small ...”
      Great Answer
      Good Answer
      Quantum Mechanics (small particles) obey rules driven by probability. Further more, this level of complexity comes in when considering fluid dynamics. Weather is driven by fluid dynamics, understanding river flows, economics, ecological systems. In fact, most "natural" or "living" systems that have been called chaotic are affected by probability.
      (26 votes)
  • piceratops ultimate style avatar for user Macias, Anthony
    Posted 6 years ago. Direct link to Macias, Anthony's post “Has anyone here ACTUALLY ...”
    Has anyone here ACTUALLY had experimental results match your theorectical probability statement?
    (43 votes)
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    • mr pants teal style avatar for user Vincent Bailey
      Posted 6 years ago. Direct link to Vincent Bailey's post “It happens quite a bit. G...”
      Great Answer
      Good Answer
      It happens quite a bit. Go pick up a coin and flip it twice, checking for heads. Your theoretical probability statement would be Pr[H] = .5.

      More than likely, you're going to get 1 out of 2 to be heads. That would be very feasible example of experimental probability matching theoretical probability.
      (40 votes)
  • leafers ultimate style avatar for user keeganrossland
    Posted 5 years ago. Direct link to keeganrossland's post “Just asking, does anyone ...”
    Just asking, does anyone know how this can relate to everyday life?
    (I'm just curious) I don't see it myself so I'm looking if someone does.
    (6 votes)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user kstovall
      Posted 5 years ago. Direct link to kstovall's post “At a restaurant, they use...”
      Great Answer
      Good Answer
      At a restaurant, they use probability in an attempt to figure out how many guests will come in and dine on any given day. At the restaurant I work at, they use the amount of reservations for the day, and statistics from the past to come up with this probability. The amount of guests that might come in and dine will determine how many servers they will need to, and how many kitchen staff will be needed to provide for all of the guests. Every night their are servers "on call", and if the probability that "a lot" (my words, but their predictions are more accurate) of guests will come in means that they will call in more servers. And if "not a lot" a guests are expected to come in, then more servers (and kitchen staff) will not be called in, or some servers might be offered to take the night off.
      (39 votes)
  • aqualine ultimate style avatar for user Emmanuel Morgan
    Posted 6 years ago. Direct link to Emmanuel Morgan's post “how do you answer questio...”
    how do you answer question number PART 2
    (5 votes)
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    • duskpin tree style avatar for user shayalipatel
      Posted 6 years ago. Direct link to shayalipatel's post “There are six faces on a ...”
      Good Answer
      There are six faces on a die and each face has the same probability (1 ∕ 6) of coming up whenever we roll the die.
      This means that if we roll a die 60 times we can expect each of the six faces to come up an equal amount of times, which would be 60 ∕ 6 = 10.
      Thereby we can say that as Dave is to roll a die 60 times, he should expect 10 of those rolls to show a 1.
      (15 votes)
  • duskpin sapling style avatar for user Elli Michalopoulou
    Posted 5 years ago. Direct link to Elli Michalopoulou's post “In QUESTION B, I don't un...”
    In QUESTION B, I don't understand where the 60 comes from
    (8 votes)
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  • blobby green style avatar for user molviwaleed
    Posted 5 years ago. Direct link to molviwaleed's post “can any one answer this q...”
    can any one answer this question please. i need to understand this
    a sensitive piece of electronic gear has a switch that fails on the average once every 100 missions engineers have devised a test which indicates defective 90 percent of the time if the switch is defective and non defective 99 percent of the time if it is okay the gear has failed, and it is found that the witch was tested and found defective. what is the probability that switch was indeed defective?
    (2 votes)
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    • duskpin tree style avatar for user vivi
      Posted 5 months ago. Direct link to vivi's post “This is conditional proba...”
      This is conditional probability.
      The formula is P(A|B)(what you are trying to find) = P(A and B)/P(B)
      With this question you want to find the probability that the switch is actually defective given that the test shows it as defective.

      So you would need to find the probability the switch is actually defective AND the test shows it as defective. Then you divide it by the probability that the test shows it is defective.

      The the probability the switch is actually defective AND the test shows it as defective is (.01)(.9) or .009

      And the probability the test shows it as defective is (.01)(.9)+(.99)(.01)=.0189

      Now you divide and you get 0.47619048 which is the probability the switch is actually defective given the test shows it is defective.

      If this is hard to understand, try making a tree diagram.

      Also anyone feel free to correct me if I am wrong, I am pretty sure I am right but I have only had one cup of coffee.
      (1 vote)
  • winston baby style avatar for user Saranya
    Posted 5 years ago. Direct link to Saranya's post “So in experimental probab...”
    So in experimental probability, you're just finding the probability of something showing up from the experiments, and theoretical is what should actually happen? Also, are the equations the same or different?
    (7 votes)
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  • blobby green style avatar for user sudsy zutphin
    Posted 4 years ago. Direct link to sudsy zutphin's post “I'm researching the proba...”
    I'm researching the probability/likelyhood that 2 or more of our local morning news casts (5 different TV broadcasting stations) would duplicate a story/topic. The average number of stories in a 30 minute show for all stations is ~ 10. The average number of duplicate stories over a 6 month period varies -- my semi-scientific tracking indicates a range of 3-5 are the same topic on 2 or more stations. I sense this is high.

    I live in a state that encompasses 104K sq mi with 5.5 million residents. The 'fact' that 30-50% of the stories are duplicated across 2 - 5 channels gives me a sense that 1) there are a good deal of inactive reporters 2) channels share some type of news feed from which they select stories or 3) there's some sort of subjective selection occurring.. (the channels are fox, kwgn, abc, nbc and cbs affiliates)

    Would you be so kind as to identify the first couple of steps of the approach I might use to begin to narrow down a legitimate estimated probability of duplicate stories based on the small amount of data I have collected?
    Regards,
    Sudsy
    (5 votes)
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  • hopper cool style avatar for user SirSergeant
    Posted 5 years ago. Direct link to SirSergeant's post “That you very much, I hav...”
    That you very much, I have learned the difference between the two probabilities in this, the videos, and the practices
    (5 votes)
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  • ohnoes default style avatar for user MysticalMagic
    Posted 2 years ago. Direct link to MysticalMagic's post “Well, It is possible to g...”
    Well, It is possible to get a million heads in a row, its just really REALLY REALLY REALLY REALLY REALLY REALLY REALLY unlikely, I calculated on Wolfram alpha and it was 1 in 10^10^5.4 aproximately. thats 301000 digits, but I guess anything is possible, right?
    (3 votes)
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