Austin and Sara s Game

Transcription

1 Austin and Sara s Game 1. Suppose Austin picks a random whole number from 1 to 5 twice and adds them together. And suppose Sara picks a random whole number from 1 to 10. High score wins. What would you guess is the proportion of times Austin would win, if this game were played many times? What would you guess is the proportion of times Sara would win? 2. Play a few rounds of the game using the digits from line 101 in the random digits table (found below) in the order that they appear. Read the digit 0 as the number 10. For example, the first block of digits is Austin picks two numbers first. His numbers are 1 and 2. Note that we ignore the 9 and go on to the next digit because the numbers he chooses have to be in the 1 to 5 range. Then the next digit (the second 2) is Sara s number. Since Austin s 1+2=3 is greater than Sara s 2, Austin wins the first game. Record the results. Continue in this fashion, reading left to right in row 101 and discarding digits as necessary, until you come to the end of the row. At this point, is either player pulling ahead, or are they about even? Line 101: Carefully enter the following commands in the home screen on your calculator. randint(1,5,200) L1: randint(1,5,200) L2: randint(1,10,200) L3: ((L1+L2) > L3) L4: sum(l4) / 200. The randint command is found under MATH/PRB/5: randint. The symbol is the STO> key. The : symbol is found by ALPHA/(decimal point). The > symbol can be found under 2nd/MATH/TEST/3: >. The sum command is found under 2nd/STAT/MATH/5: sum(. This sequence of commands tells the calculator to pretend to play the game 200 times and record the number of times Austin wins. Press ENTER to carry out the simulation, and record the results. Are you surprised? Press ENTER again to simulate 200 more games, and record the results. Continue to press ENTER until you have 25 sets of repetitions for a total of 5,000 games.

2 4. Enter your 25 results in list L5. Create a histogram of this data under 2nd/STAT PLOT/Plot 1. Define the window as: Xmin = 0.4, Xmax = 0.6, Xscl = Ymin = -4, Ymax = 20, Yscl = 0.1 Then create a boxplot under 2nd/STAT PLOT/Plot 2. Press GRAPH to see both the histogram and boxplot. TRACE to find the center of both plots. Based on what you see, what is your prediction for the proportion of times Austin would win if this game were played many times? 5. Do 1 Var Stats and look at the mean and median for your data. Do they tend to agree with your prediction in Step 4? 6. Now let s consider the game from Sara s perspective. Do you think that Sara s chances of winning over the long terms are the same as Austin s? What minor change to the calculator instructions in Step 3 would simulate 200 plays of the game and report the proportion of times Sara wins? Make this change, and then press ENTER repeatedly until you get 25 results. Calculate the mean of the 25 decimals. Are you surprised? 7. Would you like to modify your estimate of the proportion of times Sara wins? If so, what is your new estimate?

3 Probability Models The SAMPLE SPACE (usually represented with an S ) of a situation involving chance is the set of all the possible outcomes of that situation. For example the sample space of rolling a die is {1, 2, 3, 4, 5, 6}. (The use of bracket symbols around the sample space is called set notation.) Each member of S is a possible sample from the die, which is where the term sample space comes from. 1. Suppose that you flip a coin and then roll a die. How many outcomes are there in the sample space? Write the sample space using set notation. 2. How many outcomes are there in the sample space of an experiment consisting of flipping three coins? Write the sample space using set notation in two ways: 1) If order matters 2) If order doesn t matter 3. If a combination lock has a three-number combination and the wheel on the lock allows any number from 0 to 39, how many outcomes are in the sample space of possible combinations? (You probably don t want to list out all the different combinations.)

4 Conditional Probability 1. The following two-way table gives the number of students at George Washington High School participating in different after-school activities by grade. 9 th 10 th 11 th 12 th Total Athletics Fine Arts Other Total a) What is the probability a randomly selected student is an athlete? (Express your answer as a fraction.) P(athlete) = b) What is the probability a randomly selected student is an athlete and a 10 th grader? (Express your answer as a fraction.) P(athlete 10 th grader) = c) Find the probability that a randomly selected student is a 10 th grader, given that he/she is an athlete. This is written as P(10 th grader athlete). You read the bar as given. P(10 th grader athlete) = 2. Two different airlines have a flight from Los Angeles to New York that departs each weekday morning at a certain time. The probability that the first airline s flight is fully booked on a particular day is.6 and the probability that the second airline s flight is fully booked is.7. The probability that both airlines flights are fully booked is.54. a) The probability that both airlines flights are fully booked is.54, instead of.6.7 =.42. What conclusion can you draw as a result? b) Calculate the probability that the second airline s flight will be fully booked given that the first airline s flight is fully booked. c) Does the probability that the second airline s flight will be fully booked increase, decrease, or stay the same if we know that the first airline s flight is fully booked?

5 Probability Rules Part 1 1. Shade the event in the Venn diagram provided. a) E F ( E intersect F ) shade b) E F ( E union F ) shade everything in the rectangle that is in everything in the rectangle that is in E and F E or F E F E F c) E C (complement of E) shade d) only E shade everything in the everything in the rectangle that is not in rectangle that is only in E E E F E F SHOW YOUR WORK ON THE PROBLEMS THAT FOLLOW SO THAT YOUR NOTES WILL BE USEFUL WHEN YOU LOOK AT THEM LATER. 2. A radio station that plays classical music has a by request program each Saturday evening. The percentages of requests for composers on a particular night are as follows: Bach 5% Mozart 21% Beethoven 26% Schubert 12% Brahms 9% Schumann 7% Dvorak 2% Tchaikovsky?% Mendelssohn 3% Wagner 1% Suppose that these are the only composers requested and one of these requests is randomly selected. a) What is the probability that the request is for Tchaikovsky?

6 b) Find P(Dvorak Mendelssohn). (In words, what is the probability that the request is for Dvorak or Mendelssohn?) Make and label a Venn diagram that represents this situation. c) Find P(Mozart C ). (In words, what is the probability that the request is not for Mozart?) Make and label a Venn diagram that represents this situation. d) What is the probability that the request is for one of the three composers whose last name begins with B? e) What is the probability that the request is not for one of the two composers whose last name begins with S? f) Bach and Wagner are the only composers in the list above who never wrote any symphonies. What is the probability that the request is for a composer who wrote at least one symphony? A PROBABILITY MODEL for a situation involving chance consists of (1) a list of all the possible outcomes and (2) the probability of each possible outcome. 3. a) Give the probability model for rolling a six-sided die one time. Outcome Probability b) If we roll two six-sided dice, what is probability that the sum of the dice is 6? c) What is the probability that the sum of two six-sided dice will not be 6? d) What is the probability that the sum of two six-sided dice will be 5 or 6?

7 Probability Rules Part 2 SHOW YOUR WORK ON THE PROBLEMS THAT FOLLOW SO THAT YOUR NOTES WILL BE USEFUL WHEN YOU LOOK AT THEM LATER. 1. Suppose that you flip a coin and then roll a die. a) What is the probability that the coin will come up heads? b) What is the probability that you roll a number higher than 4? c) What is the probability that the coin comes up heads and the die comes up with a number higher than 4? (You may want to look back at your work from a couple days ago where you listed the combinations of possible outcomes. Also, make sure you see how your answer is related to the answers for the other parts of this problem.)

8 2. Approximately 15% of those eligible are called for jury duty in any one calendar year. People are selected for jury duty at random from those eligible. The same individual cannot be called more than once in the same year, but are eligible to be selected again the next year. a) What is the probability that a particular eligible person is not selected one year? b) What is the probability that a particular eligible person is selected three years in a row? c) What is the probability that a particular eligible person is not selected ten years in a row? d) What is the probability that a particular eligible person is selected two years in a row and is not selected the third year? e) What is the probability that a particular eligible person is selected exactly one time in a two-year period? Make and label a tree-diagram that represents the situation.

9 Probability Rules Part 3 SHOW YOUR WORK ON THE PROBLEMS THAT FOLLOW SO THAT YOUR NOTES WILL BE USEFUL WHEN YOU LOOK AT THEM LATER. 1. Suppose that you roll a die one time. a) What is the probability of rolling a 1? b) What is the probability of rolling a 2? c) What is the probability of rolling a 3? d) What is the probability of rolling a 1, 2, or 3? Make and label a Venn diagram that represents the situation. 2. Suppose that you draw one card from a standard deck of 52 cards. The probability of drawing a red card is 26/52, the probability of drawing an ace is 4/52, and the probability of drawing a red ace is 2/52. Use these probabilities to find the probability of drawing an ace or a red card. Make and label a Venn diagram that represents the situation. 3. There are two traffic lights on the route used by a certain individual to go from home to work. Let E denote the event that the individual must stop at the first light, and define the event F in a similar manner for the second light. Suppose that P(E) =.4, P(F) =.3, and P(E F) =.15. What is the probability that the individual must stop at least one light; that is, what is the probability of E F? Make and label a Venn diagram that represents the situation.

10 4. In 1998 the American Film Institute created a list of the top 100 films ever made ( Suppose that the probability that Allen, Beth, and Frank have all seen a randomly selected film from the list is 39% (i.e. all three of them have seen 39 of the 100 films) the probability that Allen and Beth have both seen a randomly selected film from the list is 42% the probability that Allen and Frank have both seen a randomly selected film from the list is also 42% the probability that Frank will be the only one who has seen a randomly selected film is 6% the probability that Allen has seen a randomly selected film from the list is 48% the probability that Beth has seen a randomly selected film from the list is 59% the probability that Frank has seen a randomly selected film from the list is 62% a) Record the probabilities given above in the Venn diagram. The first probability is easy be careful with the rest of them. Allen Beth Frank b) What is the probability only Allen has seen a randomly selected film from the list? c) What is the probability that Beth and Frank have both seen a randomly selected film from the list that Allen has not seen? d) What is the probability that neither Allen nor Beth nor Frank has seen a randomly selected film from the list?

11 3. We will find the probability of getting all Aces when four cards are dealt. a) What is the probability that the first card is an Ace? b) What is the probability that the second card is an Ace if the first card was an Ace too? Is this event independent from the event in part a? c) What is the probability that the third card is an Ace if the first two cards were Aces too? Is this event independent from the event in part a and b? d) What is the probability that the fourth card is an Ace if the first three cards were Aces too? Is this event independent from the event in part a, b, and c? e) What is the probability that all four cards will be Aces? 4. According to life tables used by the National Center for Health Statistics, the proportion of black males who will live to age 20 is.947. The proportion who live to age 65 is.499. On the basis of these proportions, find the probability that a black male who is 20 years old will reach age 65.

12 AP Stats Conditional Probability Homework

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14 More on Conditional Probability 1. If P(B) =.5, P(B A) =.5, are A and B independent? Explain. 2. The probability of having a certain disease is.05. The probability of testing positive if you have the disease is.98; the probability of testing positive when you do not have the disease is.10. a. Draw a tree diagram in which the first stage is whether or not you have the disease and the second is the outcome of the test. Include the probability of each segment in your diagram. b. P(have the disease and test positive) =? c. P(test positive) =? d. P(have the disease test positive) =? (over)

15 3. John has coronary artery disease. If he chooses to have bypass surgery, there is probability 0.05 that John will not survive, probability 0.10 that he will survive with serious complications, and probability 0.85 that he will with no complications. If he survives with complications, the conditional probability that he survives for 5 years is If there are no serious complications, the conditional probability that he survives for 5 years is a. Draw a tree diagram that summarizes the information. b. Calculate the probability that John survives for 5 years assuming that he chooses surgery. c. The other option, medical management of the coronary artery disease, gives John a 0.7 probability of surviving for 5 years. Does surgery or medical management offer John a better chance of surviving for 5 years?

16 4. Suppose automobile license plates consist of three letters followed by three digits. Determine the number of outcomes in the sample space of license plates. 5. If duplicate letters and duplicate digits are not allowed in the same situation as #4, what is the number of outcomes in the sample space? 6. A bag contains 15 cards with the numbers 1 through 10 printed on them. Each even number appears on one card and each odd number appears on two cards. One card is to be drawn and the number recorded. How many outcomes are there in the sample space? Write the sample space using set notation. 7. How many outcomes are there in the sample space if two cards are drawn, if the first card is replaced before the second card is drawn? 8. How many outcomes are there in the sample space if two cards are drawn, if the first card is not replaced before the second card is drawn?