Math 1313 Section 6.2 Definition of Probability

Transcription

1 Math 1313 Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability that it will rain is Experiment: Toss a coin. Sample Space: {H, T}. If the coin is unbiased, the chances of getting heads is 50%. 1 That is, the probability of tossing a head is P ( H ). 2 Another way to find P(H) is based on continued experimentation. A coin is tossed several times and the results are recorded as below: As the number of trials becomes very large, the relative frequency approaches 0.5. That is, P(H)=1/2. Consider an experiment that can be repeated over and over again under independent and similar conditions. Suppose that in n trials the event E occurs m times. m The ratio is called the relative frequency of an event E after n repetitions. n If this frequency approaches a value P(E) as n becomes larger and larger, then P(E) is called the probability of the event E. 1

2 Here is some general information about probability: The probability that an event will occur is a number between 0 and 1, inclusive. 0% chance of happening probability is 0. 5% chance of happening probability is 5/100 = % chance of happening probability if 20/100= % chance of happening probability is % chance of happening (will happen for sure) probability is 1. An event that cannot occur has probability 0. An event that is certain to occur has probability 1. The more likely an event is to occur, the closer the probability will be to 1. The less likely an event is to occur, the closer the probability will be to 0. Example: There are 2 black cards in a box. Chose one card randomly. What is the probability that the card is black? What is the probability that the card is red? Answer: Answer: Example: The probability of an event E is 0.75 and another event F is 0.2. Which one is more likely to happen? 2

3 If S { s1, s2,..., s n } is a finite sample space with n outcomes, then the events { s },{ s },...{ s } are called simple events. 1 2 n Simple events are mutually exclusive. The probabilities of all of the outcomes in the sample space will add up to 1. Example: Toss a coin. Simple events: {H}, {T} Probability of {H} = 1/2 Probability of {T} = 1/2 Example: Roll a die Simple events: {1}, {2}, {3}, {4}, {5}, {6} Probability of {1} = 1/6 Probability of {2} = 1/6, Probability of {3} = 1/6, Probability of {4} = 1/6, Probability of {5} = 1/6, Probability of {6} = 1/6. 3

4 Example 1: A box contains 2 red, 5 blue and 6 white cards. One card is drawn from the box. List the simple events. Find the probability distribution associated with this experiment. Simple event Probability 4

5 Example 2: The accompanying data were obtained from a survey of 100 Americans who were asked: How safe are American-made consumer products? Rating Number of Respondents A (Very Safe) 24 B (Somewhat Safe) 41 C (Not too Safe) 20 D (Not Safe at All) 10 E (Don t Know) 5 Find the probability distribution associated with this experiment. Simple event Probability 5

6 Definition: A sample space in which the outcomes of an experiment are equally likely to occur is called a uniform sample space. Let S { s1, s2,..., s n } be a uniform sample space. Then P s ) P( s )... P( s ) ( 1 2 n 1 n Example: A fair die is cast. Is this sample space uniform? Simple events: {1}, {2}, {3}, {4}, {5}, {6} Probability of {1} = 1/6 Probability of {2} = 1/6, Probability of {3} = 1/6, Probability of {4} = 1/6, Probability of {5} = 1/6, Probability of {6} = 1/6, Example: A box has 2 red cards and 1 black card. One card is chosen randomly. Is this a uniform sample space? Simple event Probability {Red} 2/3 {Black} 1/3 6

7 Finding the Probability of an event E If an event includes 2 or more members in the sample space, we can find the probability of the event by adding the probabilities of the simple events together. For example, if E { s1, s2} ; then the probability of event E is: P( E) P( s ) P( s ) 1 2 Example 3: The probability distribution for the teams A, B and C: Team Probability of winning A 0.20 B 0.55 C 0.25 What is the probability that team A or B will win? 7

8 Example 4: A fair dice is cast. What is the probability of getting an odd number? the number of desired outcomes OR: PE ( ) the number of all outcomes 8

9 Example 5: There are 200 students in a class. A survey showed that 80 students like math. What is the probability that a person chosen at random from this class likes math? Example 6: There are 8 blue and 7 green marbles in an urn. A marble is selected at random. What is the probability that the marble is blue? 9

10 Example 7: A box contains 5 cards numbered 2, 4, 6, 8, 10. What is the probability that a card chosen at random has number 2? What is the probability that a card chosen at random has a number less than 7? What is the probability that a card chosen at random has an even number? What is the probability that a card chosen at random has an odd number? 10

11 Example 8: A pair of fair dice is cast. a. What is the probability of getting two 6s? b. What is the probability that the sum is odd? c. What is the probability that at least one 2 is cast? d. What is the probability that both dice show the same number? 11

12 Example 9: If one card is drawn from a well-shuffled standard 52-card deck, what is the probability that the card drawn is a. a club? b. a red card? c. a seven? d. the ace of spades? 12

13 Popper for Section 6.2 Question# : A box contains 4 red and 5 blue cards. One card is chosen randomly. What is the probability that the card is red? A) 4/5 B) 5/9 C) 4/9 D) 5/4 E) 9/4 F) None of these 13