Probability Rules. 2) The probability, P, of any event ranges from which of the following?

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1 Name: WORKSHEET : Date: Answer the following questions. 1) Probability of event E occurring is... P(E) = Number of ways to get E/Total number of outcomes possible in S, the sample space....if. 2) The probability, P, of any event ranges from which of the following? 3) If E c represents the complement of event E then P(E c ) equals which of the following? 4) The probability distribution for an experiment is a table or chart giving the probability associated with all the unique events in an experiment. There are 2 criteria required for the probability distribution. 1. Each event in the distribution must be mutually exclusive (no overlapping outcomes). 2. The sum of probabilities of all events must be 1. What is the probability distribution of an experiment to calculate the sum of two fair, sixsided dice when rolled? 5) The sample space from an experimental roll of two dice includes how many outcomes? 6) Tree diagrams are a useful tool to directly compute the. 7) If an experiment is designed to find the sum of the rolls of two fair, six-sided dice then how many compound events result? 8) If event E is mutually exclusive with event F (no overlapping outcomes), then the probability of event E or event F, P(E F), occurring is equal to which of the following?

2 WORKSHEET: Solve the following problems. 1) If the probability of Event A is 25%, then what is the probability of not Event A? 2) If the letters to the word PROBABILITY are placed in a bag and randomly selected, what is the probability of not selecting a B on the first draw? 3) Two fair dice are thrown. What is the probability that the sum does not add to 4? 4) What is the probability of getting all heads on three coin flips in a row? 5) What is the probability of getting at least one head on three coin flips in a row? 6) Jan selects marbles randomly from a bag that contains only 40 white marbles, 24 green marbles, and 16 blue marbles. If he picks 10 marbles, how many marbles of each color did Jan most likely pick? 7) A bag contains only blue marbles and green marbles. Sixty of the marbles are green. If a marble is randomly drawn from the bag, there is a 60% chance that it will be blue. How many blue marbles are in the bag? 8) The probability of drawing a green candy from a jar of 20 candies is 1/4. How many yellow candies should be added to the jar in order to reduce the probability to 1/6? 9) A box contains 19 marbles 9 blue and 10 white. Seven of these marbles are removed at random. If the probability of drawing a blue marble is now 1/3, how many white marbles were removed from the box? 10) Of 27 marbles in a can, 7 were black, 4 were yellow, and the rest were red. Jay removed 3 black marbles, then one more marble at random. What is the probability that it was red?

3 ANSWERS : 1) Probability of event E occurring is... P(E) = Number of ways to get E/Total number of outcomes possible in S, the sample space....if there is a uniform sample space where every outcome is equally likely. 2) The probability, P, of any event ranges from which of the following? 0 P(E) 1 3) If E c represents the complement of event E then P(E c ) equals which of the following? 1 - P(E) 4) The probability distribution for an experiment is a table or chart giving the probability associated with all the unique events in an experiment. There are 2 criteria required for the probability distribution. 1. Each event in the distribution must be mutually exclusive (no overlapping outcomes). 2. The sum of probabilities of all events must be 1. What is the probability distribution of an experiment to calculate the sum of two fair, sixsided dice when rolled? 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 5) The sample space from an experimental roll of two dice includes how many outcomes? 36 6) Tree diagrams are a useful tool to directly compute the sample space. 7) If an experiment is designed to find the sum of the rolls of two fair, six-sided dice then how many compound events result? 11 (see answer to #4) 8) If event E is mutually exclusive with event F (no overlapping outcomes), then the probability of event E or event F, P(E F), occurring is equal to which of the following? P(E) + P(F)

4 ANSWERS : Solve the following problems. 1) If the probability of Event A is 25%, then what is the probability of not Event A? 75% 2) If the letters to the word PROBABILITY are placed in a bag and randomly selected, what is the probability of not selecting a B on the first draw? 9/11 3) Two fair dice are thrown. What is the probability that the sum does not add to 4? 11/12 4) What is the probability of getting all heads on three coin flips in a row? 1/8 5) What is the probability of getting at least one head on three coin flips in a row? 7/8 6) Jan selects marbles randomly from a bag that contains only 40 white marbles, 24 green marbles, and 16 blue marbles. If he picks 10 marbles, how many marbles of each color did Jan most likely pick? 5 white, 3 green and 2 blue 7) A bag contains only blue marbles and green marbles. Sixty of the marbles are green. If a marble is randomly drawn from the bag, there is a 60% chance that it will be blue. How many blue marbles are in the bag? 90 8) The probability of drawing a green candy from a jar of 20 candies is 1/4. How many yellow candies should be added to the jar in order to reduce the probability to 1/6? 10 9) A box contains 19 marbles 9 blue and 10 white. Seven of these marbles are removed at random. If the probability of drawing a blue marble is now 1/3, how many white marbles were removed from the box? 2 10) Of 27 marbles in a can, 7 were black, 4 were yellow, and the rest were red. Jay removed 3 black marbles, then one more marble at random. What is the probability that it was red? 2/3

5 KEY CONCEPTS:. 1. Sample Spaces & Events - An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment is called an outcome of the experiment. The sample space of an experiment is the set of all possible outcomes. It is important to keep in mind what is being observed or recorded in the experiment. Example: Determine the sample space, S, for the following experiments. Flipping a coin and observing whether it lands heads or tails. Rolling a fair die and observing the number that is rolled. Rolling two fair dice and observing the sum of the numbers rolled. An event is a subset of the sample space of an experiment. An elementary (or simple) event is an eventthat consists of a single outcome. Example: Consider the experiment of rolling two fair dice and observing the numbers that are rolled on each die. The sample space S for this experiment is S = (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6) The first coordinate of these ordered pairs represents the first die and the second coordinate represents the second die. (3, 5) and (5, 3) are different outcomes because in (3, 5), the first die rolls a 3, but in (5, 3) the second die rolls a 3. If it helps, think of one die as red and the other as green. 2. Probability Basics - Definition: A sample space S in which all outcomes are equally likely is called a uniform sample space. If S is a finite uniform sample space and E is any event, then the probability of E, P(E), is given by: P(E) = Number of ways for E to occur / Total number of possible outcomes in S = n(e) / n(s)

6 Note: Probabilities will ALWAYS be between 0 and 1, inclusive. The larger the probability, the more likely it is to occur. Example: Suppose a fair die is rolled and the number that lands up is recorded. The sample space for this experiment is S = {1, 2, 3, 4, 5, 6}. Sometimes experiments are run to help estimate the probability of certain events. Probabilities that are based on collected data are called empirical probabilities. If an experiment is performed n times and an event E occurs m times, then the relative frequency of the event E is m/n P(E) 1 for any event E in a sample space S. In particular P( ) = 0 and P(S) = If E and F are mutually exclusive events, then P(E F) = P(E) + P(F) 3. Union rule for probability: If E and F are ANY two events (not necessarily mutually exclusive), then P(E F) = P(E) + P(F) P(E F) Note: This formula is consistent with (2) because if two events are mutually exclusive, then E F = and thus P(E F) = P( ) = Complement Principle: P(E c ) = 1 P(E) or P(E) = 1 P(E c )