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12.3 An Introduction to Probability What is probability? What is the difference between theoretical probability and experimental probability? What is geometric probability?

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T HEORETICAL AND E XPERIMENTAL P ROBABILITY The probability of an event is a number between 0 and 1 that indicates the likelihood the event will occur. There are two types of probability: theoretical and experimental.

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T HEORETICAL AND E XPERIMENTAL P ROBABILITY THE THEORETICAL PROBABILITY OF AN EVENT When all outcomes are equally likely, the theoretical probability that an event A will occur is: P (A) = total number of outcomes The theoretical probability of an event is often simply called the probability of the event. all possible outcomes number of outcomes in A outcomes in event A outcomes in event A You can express a probability as a fraction, a decimal, or a percent. For example:, 0.5, or 50%. 1 2 P (A) = 4 9

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Finding Probabilities of Events You roll a six-sided die whose sides are numbered from 1 through 6. Find the probability of rolling a 4. S OLUTION Only one outcome corresponds to rolling a 4. P (rolling a 4) = number of ways to roll a 4 number of ways to roll the die 1 6 =

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Finding Probabilities of Events Three outcomes correspond to rolling an odd number: rolling a 1, 3, or a 5. P (rolling odd number) = number of ways to roll an odd number number of ways to roll the die You roll a six-sided die whose sides are numbered from 1 through 6. Find the probability of rolling an odd number. S OLUTION =

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Finding Probabilities of Events All six outcomes correspond to rolling a number less than 7. P (rolling less than 7 ) = number of ways to roll less than 7 You roll a six-sided die whose sides are numbered from 1 through 6. Find the probability of rolling a number less than 7. S OLUTION number of ways to roll the die 6 6 == 1

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Probabilities Involving Permutations or Combinations You put a CD that has 8 songs in your CD player. You set the player to play the songs at random. The player plays all 8 songs without repeating any song. What is the probability that the songs are played in the same order they are listed on the CD? S OLUTION There are 8! different permutations of the 8 songs. Of these, only 1 is the order in which the songs are listed on the CD. So, the probability is: Help 1 8! 1 40, 320 P(playing 8 in order) = =

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Probabilities Involving Permutations or Combinations You put a CD that has 8 songs in your CD player. You set the player to play the songs at random. The player plays all 8 songs without repeating any song. You have 4 favorite songs on the CD. What is the probability that 2 of your favorite songs are played first, in any order? S OLUTION There are 8 C 2 different combinations of 2 songs. Of these, 4 C 2 contain 2 of your favorite songs. So, the probability is: Help P(playing 2 favorites first) = = = C 2 8 C

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Probabilities Involving Permutations or Combinations Sometimes it is not possible or convenient to find the theoretical probability of an event. In such cases you may be able to calculate an experimental probability by performing an experiment, conducting a survey, or looking at the history of the event.

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Finding Experimental Probabilities In 1998 a survey asked Internet users for their ages. The results are shown in the bar graph.

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Finding Experimental Probabilities S OLUTION The number of people surveyed was = 12,443. Of the people surveyed, are at most 20 years old. So, the probability is: P(user is at most 20) = ,443 Find the experimental probability that a randomly selected Internet user is at most 20 years old.

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Finding Experimental Probabilities S OLUTION Find the experimental probability that a randomly selected Internet user is at least 41 years old. Given that 12,443 people were surveyed. Of the people surveyed, = 4190 are at least 41 years old. So, the probability is: P(user is at least 41) = ,443

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G EOMETRIC P ROBABILITY Some probabilities are found by calculating a ratio of two lengths, areas, or volumes. Such probabilities are called geometric probabilities.

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Using Area to Find Probability You throw a dart at the board shown. Your dart is equally likely to hit any point inside the square board. Are you more likely to get 10 points or 0 points?

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= = = Using Area to Find Probability S OLUTION P (10 points) = area of smallest circle area of entire board Are you more likely to get 10 points or 0 points? You are more likely to get 0 points. P (0 points) = area outside largest circle area of entire board 18 2 – ( 9 2 ) 18 2 = = = – –

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What is probability? A number between 0 and 1 of the likelihood that an event will occur. What is the difference between theoretical probability and experimental probability? When all outcomes are equally likely (theoretical) compared to the actual information collected by performing an experiment, conducting a survey or looking at the history of an event (experimental). What is geometric probability? Probabilities using the ratio of two lengths, areas or volumes.

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Assignment: P ,

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COMBINATIONS OF n OBJECTS TAKEN r AT A TIME Back The number of combinations of r objects taken from a group of n distinct objects is denoted by n C r and is given by: nCr =nCr = n !n ! (n – r )! r ! For instance, the number of combinations of 2 objects taken from a group of 5 objects is. 5 C 2 = 5!5! 3! 2!

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