Theoretical Probability (pp. 1 of 6) WHAT ARE THE CHANCES? Objectives: Investigate characteristics and laws of probability. Materials: Coin, six-sided die, four-color spinner divided into equal sections Situations: A. Flipping a coin All possible outcomes for the situation Desired event Number of desired events possible on one flip Probability F D P B. Spinning a four-color spinner All possible outcomes for Desired event the situation Number of desired events possible on one spin Probability F D P C. Rolling a six-sided die All possible outcomes for the situation Desired event Number of desired events possible on one roll Probability F D P
Theoretical Probability (pp. 2 of 6) 1. What is the total number of possible outcomes when flipping a coin? Spinning a four-color spinner? Rolling a six-sided die? 2. Where does the total number of possible outcomes occur in the fractional probability in each situation? 3. What is the number of desired events possible for each situation? 4. Where does this number occur in the fractional probability? 5. What is the sum of the probabilities for each situation? 6. Give an example for each situation that would result in a probability of one. 7. Give an example for each situation that would result in a probability of zero. Conclusions: 8. From observation in the activity, how would you define outcomes? 9. From observation in the activity, how would you define events? 10. From observation in the activity, how would you define probability? Answer the questions below based on deductions from the activities to identify four Laws of Probability. 11. What is the range for all probabilities? 12. What was the sum of the probabilities in the given situations?
Theoretical Probability (pp. 3 of 6) 13. What is the probability of an event that must occur? 14. What is the probability of an event that cannot possibly happen? Probability describes predictable long-run relationships between various possible outcomes of random processes. Probabilities can be expressed as simplified fractions, decimals, or percents. How does your definition of probability compare to the mathematical definition of probability? Outcomes are equally likely possibilities for a given situation. The set of all possible outcomes is called sample space. When a die is rolled, what is the total number of possible outcomes and what is the sample space? What is the number of ways (outcomes) six girls could be arranged in the six positions on a volleyball court? Explain how you determine the outcomes. Sue Ann has two boxes. One contains digits 0 through 9. The other contains the first eight letters of the alphabet. She wants to make three symbol codes that begin with a digit and end with two letters. She will select the number by drawing a digit from the digit box. She will select the letters by selecting letters from the letter box. She will replace the first letter drawn before drawing the second. How many outcomes are possible?
Theoretical Probability (pp. 4 of 6) Allen rolls a six-sided die and flips a coin. Use a tree diagram to determine the number of outcomes and the sample space. What is the difference in number of outcomes and the sample space? An event describes a subset of the sample space. It is the desired outcome in the situation. The probability of an event is represented as P(E). When rolling a die, what event is being described by P(5)?. When rolling a die, what event is being described by P(number less than 4)? Theoretical probability is determined by mathematically analyzing a situation and its possible outcomes. Theoretical probability is mathematically described as: PE ( ) number of desired events possible total number of possible outcomes When rolling a die, what is the probability of P(5)? How was the probability determined? When rolling a die, what is the probability of P(number less than 4)? How was the probability determined?
Theoretical Probability (pp. 5 of 6) Laws of Probability A probability is a number between 0 and 1, inclusive. The sum of all probabilities for a sample space of a situation is one. The probability of the remaining events in sample space is called the complement and is found by 1 P(E). The probability of an event that must occur is one. The probability of an event that cannot occur is zero. How do the laws of probability you determined from the activities compare to the mathematical Laws of Probability? Guided Practice A jar contains 20 marbles. Six of the marbles are black, four of the marbles are green, eight of the marbles are blue, and two of the marbles are yellow. Each marble is uniform in size and has equal chance of being picked out of the jar. a. What is the total number of outcomes? b. What is the list of sample space? c. P(blue) d. P(yellow) e. P(black) f. P(green) g. P(black), P(green), P(blue), or P(yellow) h. P(red) i. P(not green) j. Explain why each of the Laws of Probability is true or false in this situation.
Theoretical Probability (pp. 6 of 6) Independent Practice 1. How many locker tags can be made that satisfy the following conditions? The first position is a letter not including Q or Z. The last three positions are digits not including 0, and digits cannot be repeated. 2. Freemore High School is starting a girl s volleyball team. Eight girls have been selected for the team. Only six girls can be on the court at one time in positions 1, 2, 3, 4, 5, and 6. How many different arrangements of the girl s volleyball team are possible? 3. Fifty ping-pong balls numbered 1-50 are in a giant bowl. If one ball is selected at random, what is the probability that it will be divisible by nine? 4. A coin is flipped and a die is rolled. a. Construct a tree diagram to determine the sample space. b. What is the probability of getting heads and a 6? c. What is the probability of getting tails and a number greater than 4? d. What is the probability of not getting tails? e. What is the probability of not getting a 3? f. What is the probability of getting heads or tails and a number less than seven? g. What is the probability of getting heads or tails and a number greater than six? 5. A traffic light is green for 50 seconds, yellow for 10 seconds, and red for 40 seconds. a. P(green) in simplified fraction and decimal form b. P(yellow) in simplified fraction and decimal form c. P(red) in simplified fraction and decimal form d. P(not green) e. P(yellow) or P (red) f. What do you notice about the answers to d and e? Explain why this occurs. 6. John and Mary are planning on having a family with four children. a. Construct a tree diagram to determine the sample space of the possible arrangements in order of birth of boys and girls b. P(exactly 2 boys) c. P(at least 1 girl) d. P(no boys) e. P(all girls) f. What do you notice about the answers to d and e? Explain why this occurs. g. If birth order was not a factor, how many combinations would be possible?