Ch 4 Notes - Part 1 PDF

Student's Name: Homework Log MDM 4U1 – Gr. 12 Data Management Unit 1: Dealing With Uncertainty – An Introduction to Probability...

Student's Name: Homework Log MDM 4U1 – Gr. 12 Data Management Unit 1: Dealing With Uncertainty – An Introduction to Probability Still Questions having to ask in problems Parents Day Section Assigned Work class with … Initials 1 Review of Prerequisite Skills Handout 4.1 An introduction to simulations and 2 Pg. 209 #1-3, 5, 8, 10-14 experimental probability 3 4.2 Theoretical probability Pg. 218 #1-9, 11, 13-14, 16, 17 Pg. 218 #12 4 4.3 Finding probability using sets Pg. 228 #1-13 5 4.4 Conditional probability Pg. 235 #1-2, 4-7, 9-10 Pg. 236 #11, 15 6 4.4 Conditional probability Pg. 247 #8, 9 (try without replacement) 7 Quiz Pg. 218 #10 8 4.5 Finding probability using tree diagrams Pg. 235 #3, 12 Pg. 245 #1-6, 10, 13a 9 4.5 Finding probability using tree diagrams Worksheets Pg. 268 #2-5, 7-10 (omit # 5dfg, 10e) 10 Review Pg. 270 #1-2, 5, 7 11 Test Avg. Time Spent Per Night: Quiz #1 Mark: Quiz #2 Mark: Unit Test Mark: Student's Comments: Teacher’s Comments: Parent's Comments:  4.1 An Introduction to Simulations and Experimental Probability Some probability statements deal with events that are very personal. For example, you may predict your chances of getting a job. A measure of probability for an event like this can only be estimated, based on experience, and cannot be calculated exactly. When dealing with events like the survival rates for certain medical procedures, probability measures can be determined by looking at real data, such as the frequency of death, gathered over a period of time. It may also be helpful to design a simulation that will assist you in estimating the probability of an event. Fair game — a game is fair if all players have an equal chance of winning, or each player can expect to win or lose the same number of times in the long run Trial — one repetition of an experiment Random variable — a variable whose value corresponds to the outcome of a random event Discrete random variable — a variable that assumes a unique value for each outcome Expected value — informally, the value the average of the random variable’s values tends toward after many repetitions INVESTIGATION: THE COFFEE GAME Suppose each morning you play a simple game with a friend to determine who pays for coffee. Your friend tosses a coin and you call it. If you are right, your friend pays $2.00 ($1.00 for each cup of coffee); otherwise, you pay. A. Have a friend flip a coin while you predict the outcome. B. Your friend pays if you call the result correctly; otherwise, you pay. Record who pays $2.00 that day. C. Repeat this four more times to simulate one workweek. D. Repeat the simulation in Steps A to C nine more times. E. Fill out the graph over the simulated 10-week period. Week Wins Losses Total Cost 1 2 3 4 5 6 7 8 9 10 Each toss of the coin represents what is called a simple event; the results, heads or tails, are called outcomes. One week’s worth of coffee payments represents a compound event, which is the result of combining five coin tosses or trials, one for each weekday. The daily cost of coffee is a quantity that is the result of a random event, and is called the random variable. It has a finite number of possible values ($0.00 or $2.00), so it is called a discrete random variable. It is possible that, for any given week, either you or your friend could lose every coin toss. Nevertheless, if the game is fair, both of you should expect to win or lose half the time. Since the total cost of coffee is $2.00 a day, the expected value for your daily coffee cost is $1.00. The expected value for the week is, therefore, $5.00. 1. Based on your results, explain whether or not the coffee game represents a fair game. 2. How many trials would be needed before you could predict the expected weekly cost with confidence? Explain your answer. (Note: You need to make sure that the number of trials is large enough to permit a valid generalization.) 3. (a) Would the game be fair if, instead of tossing a coin, your partner rolled a six-sided die and you had to guess the result? Explain your answer. (b) In part (a), how much would you expect to have to pay each day? How much would your partner expect to pay? Explain your answer. Event — a set of possible outcomes of an experiment Simulation — an experiment that models an actual event Defining Probability An experiment consists of repeated trials in which a particular event and the outcomes of that event are noted. A measure of the likelihood of an event is called the probability of the event and is based on how often that event occurs in comparison with the total number of trials. The probabilities examined in this section are derived from experiments and are, therefore, called experimental probabilities. Combine the data that was gathered during Investigation 1 by each pair of students in the class. 1. How many trials does this represent? 2. Use the results to estimate the fraction of trials that resulted in (a) your winning (b) your losing 3. Use the results to estimate the following. (a) the probability that you will have to buy coffee at some point during the week (b) the probability that you won’t have to buy coffee at all during the week (c) your average or expected daily coffee cost (d) your average or expected weekly coffee cost 4. Compare your initial expected weekly coffee cost with the result obtained in part (d) above. Which result do you feel is the best indicator of what will happen when you play this game? Why? Example – Suppose a family plans to have four children. Use a simulation to estimate the likelihood that the family will have three girls in a row and then a boy. Example – Suppose that Nicola has a batting average of 0.320. This indicates 32 hits in every 100 attempts (or 8 hits in 25 attempts). Use a simulation to estimate the likelihood that this player has no hits in a game (assuming three at-bats per game). Example – Determine the experimental probability of the following: Roll Total a) rolling a 2 1 7 2 6 3 5 b) rolling an odd number 4 9 5 8 6 5 c) rolling a number less than 5  4.2 Theoretical Probability The study of probability began with the analysis of games of chance by the mathematicians Cardano, Galileo (pictured), Pascal, and Fermat. When you state the probability of an event, you are making a statement about the likelihood of that event occurring. How do you find the probability of an event without performing an experiment? GENERAL DEFINITION OF PROBABILITY The previous section introduced you to the concept of experimental probability. Actual data were used to determine the relative frequency of a particular event. Probability is often used to predict the likelihood that a particular event will occur. Experimental probabilities or relative frequencies determined from surveys only give an estimate of the likelihood that a particular event will occur. It is possible to determine a more accurate probability for some events, such as rolling a 4 on a die. For example, in rolling a die six times, 4 may show up twice, so the experimental probability 2 is. 6 However, we know that there are six possible outcomes when a die is rolled. Only one of these outcomes is the event of rolling a 4. This is an example of a simple event. Since the possible outcomes are all equally likely to happen, it is reasonable to expect that the fraction of the time you roll a 4 is the ratio of the number of ways a 4 can occur to the number of possible outcomes. 1 P(rolling a 4) = 6 The resulting value is called the theoretical probability. Given a large enough number of trials, the theoretical probability and the experimental probability should be approximately equal. simple event — an event that consists of exactly one outcome theoretical probability — when all the outcomes of an experiment that correspond to an event are equally likely, the probability of the event is the ratio of the number of outcomes that make up that event to the total number of possible outcomes sample space — the collection of all possible outcomes of the experiment event space — the collection of all outcomes of an experiment that correspond to a particular event A Venn diagram can be used to show the relationship between the event space, A, and the sample space, S. Example – A bag contains five red marbles, three blue marbles, and two white marbles. What is the probability of drawing a blue marble? Example – If a single die is rolled, determine the probability of rolling (a) an even number (b) a number greater than 2 Probability and Complementary Events 2 In the last example, P(number > 2) =. What is P(number ≤ 2)? 3 The event of a number ≤ 2 contains all outcomes in the sample space that are not in A number > 2. These numbers are in the complement of A and are represented by A′. The Venn diagram shows A′ as the shaded region within S that is entirely outside of A. Example – A standard deck of cards comprises 52 cards in four suits—clubs, hearts, diamonds, and spades. Each suit consists of 13 cards—ace through 10, jack, queen, and king. What is the probability of drawing an ace from a well-shuffled deck? What is the probability of drawing anything but an ace?  4.3 Finding Probability Using Sets COUNTING OUTCOMES WITH VENN DIAGRAMS A Venn diagram — named after John Venn (1834-1923), a British priest and logician — can be used to graphically describe the relationships between possible results of an experiment (or survey). Compound events are shown as combinations of simpler events. All the events exist within the larger collection of all possible outcomes of the experiment. This large set is called the sample space for the experiment. The letter S commonly represents the sample space. Venn diagram — a diagram in which sets are represented by shaded or coloured geometrical shapes Compound event — consists of two or more simple events Subset — a set whose members are all members of another set Disjoint — two sets that have no elements in common Union — the set containing all of the elements in A as well as B Venn Diagrams and Set Terminology The following terminology and symbols are used in working with sets. The set A ∩ B is represented by the region of overlap of the two sets in the Venn diagram to the right. Sets A and B exist as sets within the larger set S. They are subsets of the set S. The Venn diagram to the right shows disjoint sets A and B. The set A ∪ B is represented by the shaded area in the Venn diagram to the right. Discussion 1. Why is n(∅′) = n(S) ? 2. Why is n(S′) = n(∅) ? 3. If A and B are disjoint, why is n( A ∪ B ) = n( A) + n( B ) ? 4. If A and B are not disjoint, why is n( A ∪ B ) < n( A) + n( B ) ? ADDITIVE PRINCIPLE FOR UNIONS OF TWO SETS The solution to the counting problem above employed a Venn diagram. The counting strategy that was used leads to a more general counting strategy for unions of sets, called the Additive Principle for unions of two sets. Example – 20 people are having fruit. 11 people are eating an apple. 8 people are eating a banana. 3 people are eating both. a) How many people are eating an apple or a banana? b) How many people are eating neither? Mutually exclusive events — A and B are mutually exclusive events if A ∩ B = ∅ and, as a result, P( A ∪ B ) = P( A) + P( B) since P( A ∩ B ) = ∅. Example – Suppose a survey of 100 Grade 12 mathematics students in a local high school produced the following results. Example – If you were asked to randomly select a student from the group of students described in Example 1, what is the probability that (a) the student selected is enrolled only in AFIC? (b) a student was in AFIC or in Data Management? Example – If two dice are rolled, one red and one green, find the probability that a total of (a) 2 or a total of 12 will occur (b) 4 or a pair will occur  4.4 Conditional Probability It is often necessary to know the probability of an event under restricted conditions. Recall the results of a survey of 100 Grade 12 mathematics students in a local high school. Example – Determine the probability of selecting a Data Management student, but are first told that the only students from which you can choose are enrolled in AFIC. How does this additional condition affect the probability? Example – Suppose that in Vancouver the probability that a day will be both cloudy and rainy is 25%. Suppose further that 50% of all days are cloudy. Determine the probability that it will rain given it is a cloudy day in that city. Example – What is the probability of drawing two aces in a row from a well-shuffled deck of 52 playing cards? The first card drawn is not replaced.  4.5 Finding Probability Using Tree Diagrams and Outcome Tables Games of chance often involve combinations of random events. These might involve drawing one or more cards from a deck, rolling two dice, or tossing coins. How do you find the probability that a particular sequence of these events will occur? This section will show you some strategies for listing the outcomes of an experiment in an organized way so that you can find the probability of a particular compound event. TREE DIAGRAMS AND THE MULTIPLICATIVE PRINCIPLE FOR COUNTING OUTCOMES A tree diagram is often used to show how simple experiments can be combined one after the other to form more complex experiments. The successive branches of the tree each correspond to a step required to generate the possible outcomes of an experiment. The tree diagram and outcome table that correspond to tossing two coins appear below. Consider an experiment in which you first roll a six-sided die and then toss a coin. What is the probability of tossing tails and rolling an even number? The following tree diagram can be used to represent the possible results. The sample space for this experiment consists of ordered pairs of the form (d, c ) , where d is the result of a roll of the die and c is the result of the toss of a coin. The tree diagram for the experiment clearly shows the 12 possible outcomes. For each roll of the die (the first entry in the ordered pair), we have a choice of six possible outcomes. For each of these, there are two choices for the coin toss (the second entry of the ordered pair). As a result, there are 6 × 2 = 12 possible outcomes. There are three outcomes that correspond to the event of an even roll of the die followed by a 3 1 toss of tails. They are (2, T), (4, T), and (6, T). Therefore, P(even roll, tails) = , or. 12 4 In this case, the event of getting an even roll of the die and the event of getting tails on a coin toss have no influence on each another. The following Venn diagram shows the relationship between the events. INDEPENDENT AND DEPENDENT EVENTS 3 1 In the die roll/coin toss experiment, P(even roll, tails) = , or. 12 4 What is the probability of tossing the coin and getting tails if you know in advance that the die will show an even number? The rolling of the die and the tossing of the coin are independent events. Knowing the outcome of the die roll has no effect on the probability of getting tails. Consider the conditional probability P(tails | even). From the previous section, we know that P(tails ∩ even) P(tails | even) = P(even) 1 We also know that P(tails) =. Therefore, P(tails | even) = P(tails). 2 independent events — events A and B are independent if the occurrence of one event does not change the probability of the occurrence of the other event Recall that the conditional probability of event B given event A has occurred is P (B ∩ A) P (B | A) = P ( A) P (B ∩ A) If events A and B are independent, then P(B | A) = P (B ). Therefore, P(B ) = and P ( A) P (B ∩ A) = P (B ) × P ( A). Example – The Even-Odd game is a game that can be played by two players using two coins. One of the players assumes the Even role; the other, Odd. Each player tosses a coin. If the two coins show an even number of heads, Even wins. If the coins show an odd number of heads, Odd wins. A tie is declared if no heads show. Draw a tree diagram to represent the possible events. Example – Suppose you are playing a game in which you roll a red and a black six-sided die at the same time. Consider the event that the sum of the dice is 3. Is this event independent of the red die showing 1? Example – Suppose that a school’s female soccer team has a history of winning 50% of the games it plays on rainy days and 60% of the games it plays on fair days. The weather forecast for the next game day calls for an 80% chance of rain. What is the probability that the team will win the game?