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Designing Investigations to Predict Probabilities Of Events

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What is a PROBABILITY ? PROBABILITY = # ways in which to win # total possible outcomes

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Example: Toss a coin. P(heads) = ? Answer: Note: This is THEORETICAL PROBABILITY.

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Example: The regions on the spinner below have equal areas. P(number<4) =? Answer:.75 2 1 3 4

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Suppose that you need to estimate the size of a large population. You can use the capture-recapture method. Well go through the steps, showing how to use the method.

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The CAPTURE/RECAPTURE method 1) Draw out 20 cards from a deck having an unknown number of cards. Make a pencil mark on each. 2) Return the cards to the pile. Mix them thoroughly. 3) Draw a RANDOM sample of 15 cards. Count the number of marked cards.

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Estimation: To estimate the number of cards originally in the pile: Write a proportion and solve: # marked cards drawn = 20 15 x where x is the number of cards in the original deck.

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Would we have gotten a good estimate if we purposely only chose marked cards? Its important to RANDOMLY choose the SAMPLE that we use for our estimation. A type of sample that accurately represents the entire population being considered is called a SIMPLE RANDOM SAMPLE.

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SIMPLE RANDOM SAMPLES Each member of the population has an equal chance of being selected. The selection of one member does not affect the chance of selecting another. (The selections are INDEPENDENT.)

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Which of these is an example of a simple random sample? A Choose the first 100 students that you see. B Choose every 10 th name in the student directory. C Randomly choose a homeroom and then tell the teacher to select 10 students. D Randomly assign a number to each student, put the numbers in a hat, and draw 100 numbers. ANSWER: D

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REPRESENTATIVE SAMPLES: reflect the characteristics of the population; do not leave any group out; do not over-represent any group; and do not under-represent any group. OTHERWISE, THE SAMPLE IS BIASED!! We would like an UNBIASED SAMPLE.

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SURVEY: How many students drive a car to school? How many students should we ask? (larger samples give more accurate results) Could we choose all seniors for our sample? How should we choose an unbiased sample?

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SIMULATIONS: imitate the actual event; represent virtual reality; and offer a chance to collect data to see what MIGHT happen.

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A probability simulation is… a modeling technique that makes it possible to predict the likelihood of an event or outcome without conducting the real experiment. The process imitates the actual situation.

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Steps for a Simulation: 1) State the problem. 2) State the assumptions. 3) Choose a model to generate outcomes. Assign each outcome a representative. 4) Simulate many trials.

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What models can you use? Random Number Table Random Number Generator Cards Coins Dice (number cubes) Spinner

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Jermaine applied for 3 different jobs. He thinks that there is a 50% chance he will be offered each job. Describe how Jermaine could use a spinner to model this situation. Describe a sample of data from at least 5 trials. Based on your sample data, what is the probability that Jermaine will be offered exactly two of the jobs? Note: This is EXPERIMENTAL PROBABILITY

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A pharmaceutical company is testing the effectiveness of a new medicine. Scientists test the medicine on a random sample of 20 patients. Four of the patients show improvement. Based on the test, what is the probability that a patient who is randomly selected to take the medicine would show improvement? Answer: =

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NOTE: On the HSA, you do NOT need to reduce a fraction to place it in a GRID. (student-produced answer)

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A zookeeper used a simulation to predict the genders of two tiger cubs. She tossed coins with heads (H) representing a male cub and tails (T) representing a female cub. The results of her simulation are shown below. HHHTTHTHTH HHTTTHHTTT HTTTTHTTTT TTHTTHHTTH

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Based on this simulation, what is the probability that both cubs will be the same gender? Answer: =

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A basketball player expects that she has an 80% chance to score each time that she attempts a free throw. She wants to know the probability of scoring on two consecutive free throw attempts. She uses a random number generator to conduct a simulation, where the digits 0 through 7 represent a successful free throw and 8 and 9 do not. The list below shows her simulation data for 10 trials. 2164098548 1764960074 Based on this data, what is the probability of being successful on two consecutive free throw attempts? A 20% B 40% C 60% D 80% Answer: C

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THE END Created by HaHS math teachers

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EXAMPLE 1 Using Theoretical Probability Predict the number of times a coin will land heads up in 50 coin tosses. There are two equally likely outcomes.

EXAMPLE 1 Using Theoretical Probability Predict the number of times a coin will land heads up in 50 coin tosses. There are two equally likely outcomes.

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