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experiment ( observing / measuring ) outcomes = results sample space = set of all outcomes events = subset of outcomes If we collect all outcomes we are forming a sample space If we collect some of the outcomes we are forming an event Exercise 1 Flip a balanced coin once. Write the sample space. How many outcomes we have. Exercise 2 Flip a balanced coin twice. Write the sample space. How many outcomes we have. Exercise 3 Flip a balanced coin three times. (a) Write the sample space. How many outcomes we have. (b) Write the outcomes of the events: = getting at least two heads. = getting at most one head. Exercise 4 How many possible outcomes can be obtained from flipping a balanced coin five times? Exercise 5 collection of cards numbered from 1 through 20. If one card is drawn randomly, then ( 1 ) What is the sample space? ( 2 ) Write the outcomes of the following events: = the card with an even number = the card with a number divisible by 3 C = the card with a number divisible by 4 D = the card with a number divisible by 5 : union ( either or ) : intersection ( both and ) : complement ( not ) ( 3 ) Find the following outside C D C D D mutually exclusive ( disjoint ) events 2 P a g e
0 probability 1 The probability of passing trail exam = 0.4 & The chance of passing trail exam = 40% The Classical Probability Concept The probability of an event = This rule is used if all outcomes are equally likely / equal probable elected / Draw randomly Flip / Toss a balanced coin Roll a balanced die Exercise 6 The grades that 30 students have achieved in the TT final exam are given below. If a student is randomly selected, find the probability that he/she is student. grade C D F # students 5 10 7 6 2 Exercise 7 box contains 22 red, 9 white, 11 blue, 8 black balls. If three balls are selected at random, find the probability of ( 1 ) getting all red balls ( 2 ) getting either red or blue balls ( 3 ) getting neither red nor blue balls ( 4 ) getting either all white or all black balls Exercise 8 carton of 12 light bulbs includes 3 defective. If two bulbs are chosen at random, what is the probability that: ( 1 ) exactly one bulb will be defective ( 2 ) at least one bulb will be defective ( 3 ) at most one bulb will be defective ( 4 ) both bulbs will be defective Exercise 9 In 2001, there were 520 graduates from the University of ahrain, of whom 330 were females and 190 were males. There were 40 female math-students and 25 male math-students. ( 1 ) What is the probability that a randomly selected graduate is male? ( 2 ) What is the probability that a randomly selected math-student is male? ( 3 ) What is the probability that a randomly selected graduate is math-student? ( 4 ) What is the probability that a randomly selected graduate is NOT math-student? 3 P a g e
Exercise 10 die is rolled once, what is the probability that a number less than 3 will turn up? Two dice rolled once, what is the probability of getting a sum < 10? multiple-choice question in a quiz has 4 answers. If a student choose one answer at random, what is the probability that his answer is (a) correct? (b) wrong? class consists of 15 girls and 5 boys. Find the probability of selecting 2 girls and 1 boy from this class. Venn diagrams left 1 2 3 right 4 middle outside circles Exercise 11 Use the given Venn diagram to complete the following table: Event only in only in both in & either in or in neither in nor in no. 1 2 3 4 Exercise 12 Use the shown Venn diagram to find the following probabilities: ( ) ( ) 4 P a g e
Exercise 13 Given two and such that, and Draw the corresponding Venn diagram. Calculate the following probabilities: ( ) ( ) Exercise 14 Given two mutually exclusive events and such that, ( ). Draw the corresponding Venn diagram. Calculate the following probabilities: ( ) ( ) Exercise 15 Use the shown Venn diagram to find the following probability: 0.2 0.1 0.4 ( ) 5 P a g e
Probability Rules 1. 2. 3. 4. 5. 6. If and are mutually exclusive events, then 7. If and are independent events, then 8. If and are two events, then 9. conditional probability 10. 11. 6 P a g e
Exercise 16 [ addition rule of probability ] 1. Given that and. If and are independent events, then find 2. Given that, and, then find 3. Given that and. If and are mutually exclusive events, then find Exercise 17 [ with / without repetition ] The fifth-grade class consists of 16 boys and 14 girls. If one student is selected each week to assist the instructor, find the probability that a boy is selected such that (a) the same student can serve for two weeks (b) the same student cannot serve for two weeks Exercise 18 [ mutually exclusive events ] The probabilities that the serviceability of a new laser printer will be rated very difficult, difficult, avergae, easy & very easy are respectively 0.11, 0.16, 0.35, 0.28 & 0.10. Find the probability that the serviceability of the new laser printer will be (a) average OR worse (b) average OR better OR ND union intersection NOT complement 7 P a g e
Examples of Independent Events: Rolling a die several times Flipping a coin several times RECLL Drawing with replacement ( with repetition ) Exercise 19 [ independent events ] EX1: For three rolls of a balanced die, find the probability of getting three sixes. EX2: For three rolls of a balanced die, find the probability of getting no sixes. EX3: Find the probability of getting five heads in a row with a balanced coin. EX4: box contains 2 gold rings and 4 silver rings. If 3 rings are chosen with replacement, find the probability that all are silver. EX5: You have cards numbered from 1 to 10. If you draw three of them with replacement, find the probability of getting three even numbered cards. Exercise 20 [ conditional probability ] required المطلوب condition الشرط 1. Given, and Find 2. Given and Find. 8 P a g e
Exercise 21 [ contingency table ] recent survey asked 100 people if they thought women in the armed forces should be permitted to participate in combat. Here is the result of the survey; Gender Yes Y No Total Male M 25 15 40 Female 45 15 60 Total 70 30 100 If one person is selected at random, find the following probabilities: Home Work Given the following contingency table. ex Good ad Female X 17 10 Male 13 15 If one person is selected at random, find the following probabilities: 9 P a g e