PROBABILITY - ALL ABOUT LEARNING MATHEMATICS

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PROBABILITYCandidates should able to: evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events (e.g for thetotal score when two fair dice are thrown), or by calculation using permutations or combinations;use addition and multiplication of probabilities, as appropriate, in simple cases;understand the meaning of exclusive and independent events, and calculate and use conditional probabilities insimple cases, e.g. situations that can be represented by means of a tree diagram.1. IntroductionIn many situations, you may be unsure of the outcome of some activity or experiment although you know what thepossible outcomes are. An outcome is the possible results of an experiment. For example, you do not know whatnumber you will get when you roll a dice, but you do know that you will get 1,2,3,4,5 or 6. You know that if you toss acoin twice, then the possible outcomes are (H,H), (H,T), (T,H) and (T,T). A complete listing of the possible outcomes ofan experiment is called a sample space. It is usually denoted by the letter S. The list is usually written in curly brackets,{ }.Thus, the sample space for rolling the dice is {{}.}, and the sample space for tossing a coin twice isExample 1: List out the sample space when you toss a coin once.Example 2: List out the sample space when you toss 3 coins together.Example 3: List out the sample space when you toss 2 die together.Example 4: List out the sample space when you toss a die and a coin once.1

Example 5: Throw 2 dice once, find the sum of the numbers on the top faces.Example 6: Throw 2 dice once, find the difference of the numbers on the top faces.Example 7: Throw 2 dice once, find the product of the numbers on the top faces.2. Elementary ProbabilityThe probability that an event A occurs, P(A) is given byFor example, for one throw of an ordinary die, the possibility space S {1, 2, 3, 4, 5, 6} so n(S) 6.Extra notes:(i)(ii) P (S) 1(iii) P (A’) 1 – P(A) where A’ is the event that A does not occur and it is sometimes called complement of A.(iv) P (A) 0 means that event A is impossible,(v) P (A) 1 means that event A is very certain to happen.2

Example 8: When an unbiased die is thrown once,(a) Find the probability of getting an odd number.(b) Find the probability of getting a prime number.Example 9: When an unbiased coin is tossed,(a) Find the probability of getting a head.(b) Find the probability of getting a tail.Example 10: A card is drawn randomly from a deck of poker cards. Find the probability that the card(a) is a heart;(b) is an eight; and(c) is not an eight.3. Compound Event (or Combined Event)A compound event is an event that is made up of two or more events. An example of a compound event is tossing acoin and rolling a die. These are two separate events put together to make one compound event.A venn diagram is a simple presentation of the sample space S and its relation to other sets of events.Events A and B occurringIn set language, the set that contains the outcomes that are in both A and B is called the intersection of A and B. It iswritten as. To represent it on the Venn diagram, shade the overlap area of event A and event B.3

Events A or B occurringIn set language, the set contains the outcomes that are in A or B or both is called the union of A and B. It is written as. To presenton the Venn diagram, shade the whole two circles.SinceDivide by,, this becomesAs a result, it gives us the Addition rule of Probability:Example 11: A is the event of rolling a 3 on a die and B is the event of rolling an odd number. Find(a) P (A(b) P (AB); andB).Example 12: Events A and B are such thatand. Find.4

Example 13: In a class of 30 students, 4 of the 12 boys and 5 of the 18 girls are in the athletics team. A person fromthe class is chosen to be in the 100m sprint race on sports day. Find the probability that the person chosen is(a) in the athletics team;(b) male;(c) a male member of the athletics team; and(d) a male or in the athletics team.Example 14: (Complementary events) In a hospital, there are 8 doctors and 5 nurses. 7 doctors and 3 nurses arefemales. If a staff is randomly selected, what is the probability that the subject is a male or a doctor.Example 15: Events A and B are such thatFind(a)(b)Example 15: A group of 50 people was asked which of three magazines, A, B or C they read. The results showed that24 read A, 15 read B, 14 read C, 8 read both A and B, 3 read both B and C, 5 read both A and C and 2 read all 3.(a) Represent these data on a Venn diagram.5

Find the probability that a person selected at random from this group reads(b) at least 1 of the magazines;(c) only 1 of the magazines; and(d) only A.4. Mutually Exclusive EventsTwo events A and B are said to be mutually exclusive (or exclusive) if either one or the other can occur at a time. Inother words, both A and B cannot occur at the same time. For example, with one throw of a die, you cannot score atwo and a four at the same time, so the events ‘scoring a 2’ and ‘scoring a 4’ are mutually exclusive events.If A and B are mutually exclusive (), thensinceis an impossible event. There is no overlap of A and B.For mutually exclusive events sincefor combined events becomes, the addition ruleThis rule can apply to two or more mutually exclusive events. If there are n exclusive events,Example 16: In a driving race, the probability that Fernando wins is 0.3, the probability that Lewis wins is 0.2 and theprobability that Jenson wins is 0.4. Find the probability that(a) Fernando or Jenson wins;(b) Fernando or Lewis or Jenson wins; and(c) someone else wins.6

Example 17: A card is drawn from a deck of 52 playing cards. Find the probability that the card is(a) a spade or a club;(b) a space or a Queen.5. Conditional ProbabilityConditional probability is written as. It can be defined as ‘A occurs knowing that B has occurred’ providedand. We read this as the “probability of A, given B”.Rewriting this equation givesprobability.which is known as the multiplication rule ofExample 18: Given that a club card is picked at random from a pack of 52 poker cards. What is the probability that it isa picture card?Example 19: When a die is thrown, an odd number occurs, what is the probability that the number is prime?ndExample 20: A bag contains 3 red, 2 white and 3 blue balls. A ball is drawn, what is the probability that 2 ball is redstgiven that the 1 is white.7

Example 21: X and Y are two events such that(a)(b)and. Find(c)Example 22: A group of girls at a school is entered for Advanced Level Mathematics modules. Each girl takes onlymodule M1 or only module M2 or both M1 and M2. The probability that a girl is taking M2 given that she is taking M1is . The probability that a girl is taking M1 given that she is taking M2 is . Find the probability that(a) a girl selected at random is taking both M1 and M2;(b) a girl selected at random is taking only M1.6. Independents EventsIf either of the two events A and B can occur without being affected by the other, then the two events areindependent. If A and B are independent, then P(A, given B has occurred) is precisely the same as P(A) since A is notaffected by B. In other words, P(A B) P(A). It is also true that P(B A) P(B).For independent events, the multiplication rule of probabilitybecomesThis is known as the multiplication rule for independent events. It is also known as the ‘and’ rule for independentevents.So there are three conditions for A and B to be independent and any one of them may be used as a test forindependence, i.e.(i)(ii)(iii)8

The multiplication law can be extended to any number of independent eventsExtra note:It is important not to confuse the terms ‘mutually exclusive’ and ‘independent’.Mutually exclusive events are events that cannot happen together. They are usually the outcomes of oneexperiment. For example, you cannot have a result of head and tail in one toss of a coin.Independent events are events that can happen simultaneously or can be seen to happen one after the other. Forexample, it rained on Monday and Monday is a public holiday.Example 23: If a fair coin is tossed 3 times, what is the probability of getting 3 heads in a row?Example 24: An urn contains 3 red balls, 2 blue balls and 5 white balls. A ball is selected and its colour noted thenreplaced. A second ball is then selected and its colour noted. What is the probability of selecting(a) two blue balls?(b) a blue ball then a white ball?(c) a red ball then a blue ball?Method 1: Using Multiplication RuleMethod 2: Using a Tree diagramExample 25: A poll found that 46% of Bruneians suffer from great stress at least once a week. If 3 people are selectedat random, what is the probability that all 3 will say they suffer stress at least once a week.9

7. Probability treesA useful way of tackling many problems is to draw a probability tree. The method is illustrated in the followingexample.Example 26: when a person needs a minicab, it is hired from one of three firms, X, Y and Z. Of the hire, 40% are fromX, 50% are from Y and 10% are from Z. For cabs hired from X, 9% arrive late, the corresponding percentages for cabshired from firm firms Y and Z being 6% and 20% respectively. Calculate the probability that the next cab hired(a) will be from X and will not arrive late;(b) will arrive late.(c) Given that a call is made for a minicab and that it arrives late, find, to three decimal places, the probability that itcame from Y.Example 27: Events X and Y are such thatBy drawing a tree diagram, find(a)(b)Example 28: A manufacturer makes writing pens. The manufacturer employs an inspector to check the quality of hisproduct. The inspector tested a random sample of the pens from a large batch and calculated the probability of anypen being defective as 0.025.Ali buys two of the pens made by the manufacturer.(a) Calculate the probability that both pens are defective.(b) Calculate the probability that exactly one of the pens is defective.10

The probability that a girl is taking M2 given that she is taking M1 is . The probability that a girl is taking M1 given that she is taking M2 is . Find the probability that (a) a girl selected at random is taking both M1 and M2; (b) a girl sel

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