Chapter 5 : Probability. Exercise Sheet. SHilal. 1 P a g e
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2 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
3 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 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
4 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 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 Exercise 12 Use the shown Venn diagram to find the following probabilities: ( ) ( ) 4 P a g e
5 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: ( ) 5 P a g e
6 Probability Rules If and are mutually exclusive events, then 7. If and are independent events, then 8. If and are two events, then 9. conditional probability P a g e
7 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 & 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
8 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
9 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 Female Total If one person is selected at random, find the following probabilities: Home Work Given the following contingency table. ex Good ad Female X Male If one person is selected at random, find the following probabilities: 9 P a g e
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