PROBABILITY Candidates should able to: evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events (e.g for the total 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 in simple cases, e.g. situations that can be represented by means of a tree diagram. 1. Introduction In many situations, you may be unsure of the outcome of some activity or experiment although you know what the possible outcomes are. An outcome is the possible results of an experiment. For example, you do not know what number 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 a coin twice, then the possible outcomes are (H,H), (H,T), (T,H) and (T,T). A complete listing of the possible outcomes of an 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 is Example 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 Probability The probability that an event A occurs, P(A) is given by For 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 a coin 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 occurring In set language, the set that contains the outcomes that are in both A and B is called the intersection of A and B. It is written as. To represent it on the Venn diagram, shade the overlap area of event A and event B. 3
Events A or B occurring In 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 present on the Venn diagram, shade the whole two circles. Since, Divide by, this becomes As 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 (A B); and B). Example 12: Events A and B are such that and. 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 from the 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 are females. 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 that Find (a) (b) Example 15: A group of 50 people was asked which of three magazines, A, B or C they read. The results showed that 24 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 Events Two events A and B are said to be mutually exclusive (or exclusive) if either one or the other can occur at a time. In other words, both A and B cannot occur at the same time. For example, with one throw of a die, you cannot score a two 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 ( ), then since is an impossible event. There is no overlap of A and B. For mutually exclusive events since for combined events becomes, the addition rule This 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 the probability 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 Probability Conditional probability is written as. It can be defined as A occurs knowing that B has occurred provided and. We read this as the probability of A, given B. Rewriting this equation gives probability. which is known as the multiplication rule of Example 18: Given that a club card is picked at random from a pack of 52 poker cards. What is the probability that it is a picture card? Example 19: When a die is thrown, an odd number occurs, what is the probability that the number is prime? Example 20: A bag contains 3 red, 2 white and 3 blue balls. A ball is drawn, what is the probability that 2 nd ball is red given that the 1 st is white. 7
Example 21: X and Y are two events such that and. Find (a) (b) (c) Example 22: A group of girls at a school is entered for Advanced Level Mathematics modules. Each girl takes only module M1 or only module M2 or both M1 and M2. 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 selected at random is taking only M1. 6. Independents Events If either of the two events A and B can occur without being affected by the other, then the two events are independent. If A and B are independent, then P(A, given B has occurred) is precisely the same as P(A) since A is not affected 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 probability becomes This is known as the multiplication rule for independent events. It is also known as the and rule for independent events. So there are three conditions for A and B to be independent and any one of them may be used as a test for independence, i.e. (i) (ii) (iii) 8
The multiplication law can be extended to any number of independent events Extra 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 one experiment. 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. For example, 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 then replaced. 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 Rule Method 2: Using a Tree diagram Example 25: A poll found that 46% of Bruneians suffer from great stress at least once a week. If 3 people are selected at random, what is the probability that all 3 will say they suffer stress at least once a week. 9
7. Probability trees A useful way of tackling many problems is to draw a probability tree. The method is illustrated in the following example. 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 from X, 50% are from Y and 10% are from Z. For cabs hired from X, 9% arrive late, the corresponding percentages for cabs hired 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 it came from Y. Example 27: Events X and Y are such that By drawing a tree diagram, find (a) (b) Example 28: A manufacturer makes writing pens. The manufacturer employs an inspector to check the quality of his product. The inspector tested a random sample of the pens from a large batch and calculated the probability of any pen 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