Math : Probabilities


 Erik Robertson
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1 Probability EPProgram  Strisuksa School  Roiet Math : Probabilities Dr.Wattana Toutip  Department of Mathematics Khon Kaen University 200 :Wattana Toutip wattou@kku.ac.th Probability of successive events A Tree Diagram shows the probability of successive events 20.. Examples. A bag contains 5 red and 6 black marbles. Two are drawn without replacement. What is the probability that : (a) Both are red (b) They are different colors. Solution Construct the diagram as shown. The top branch corresponds to both marbles being red. Multiply the probabilities along this branch. Fig 20. (a) The Probability that both are red is The two middle branches correspond to the marbles having different colors (b) P(different colors) Four fair dice are rolled. Find the probabilities of (a) Four 6s (b) at least one 6. Solution (a) The Probability of four 6s is 6 The Probability of four 6s is (b) There will be at least one 6 unless none of the dice show a 6. This has probability The probability of at least one 6 is
2 20..2 Exercises. Two fair dice are rolled. Find the probabilities that: (a) Both are sixes (b) The total is 2 (c) The total is 7 (d) The first is greater than the second (e) A 'double' is thrown (f) At least one of the dice is a six. 2. Two card are drawn without replacement from a wellshuffled pack. Find the probabilities that : (a) The first is a heart (b) Both are heart (c) The first is a heart and the second is a spade (d) The first is a King and the second is a Queen. 3. A bag contains 5 blue are 4 green counters. Two are drawn without replacement. Find the probabilities that : (a) Both are blue (b) They are the same colour (c) There is at least one blue (d) The second is a green. 4. A sweet box contains 5 toffees, 6 liquorices and 8 chocolates. Two are drawn out. Find the probabilities that : (a) The first is a toffee and the second is a chocolate (b) At least one is a liquorice (c) Neither is a toffee. 5. Two fair dice are rolled. The score is the larger of the numbers showing. Find the probabilities that: (a) The score is (b) The score is 6 6. Two children A and B each pick at random a single digit from to 9. Find the probabilities that : (a) They pick the same number (b) A's number is larger than B's 7. To start a certain board game a die is rolled until a six is obtained. Find the probabilities that : (a) A player starts on his first roll (b) He starts on his second roll (c) He starts on his third roll (d) He has not started by his fourth roll 8. To start at darts a' double' must first be thrown. Albert has probability of throwing a 0 double, and Beatrice has probability 8. Albert throws first. Find the probabilities that : (a)both start on their first throw (b) Beatrice starts on her second throw but Albert has not started by then. 9. A fair coin is spun five times. Find the probabilities of (a) five Heads (b) at least one Head. 0. A roulette wheel has the number to. A gambler bets that a number divisible by 3 will turn up. The bet is repeated four times. Find the probabilities that the gambler (a) Wins all his bets (b) wins at least one bets.
3 . Five people take the driving test. Each has probability of passing. Find the probabilities that : (a) They all pass (b) at least one fails Exclusive and independent events. Conditional probability If two events cannot happen together, then they are exclusive. If events are exclusive then the probability that one or the other occurs is the sum of their probabilities. P A or B P A P B, provided that A and B are exclusive. Two events are independent if the truth of the one of them does not alter the probability of the other. If events A and B are independent then the probability of them both occurring is the product of their probabilities. PA & B PA PB The conditional probability of 6! 720 A given B is the probability of A, once it is known that B is true. The conditional probability of A given B is written PA B. It is obtained from the formula : PA B PB If A and B are independent then PA B PA P A & B The symbols and are sometimes used for 'and' and 'or' respectively Example. Two cards are drawn without replacement from a pack. Events A, B, C are as follows : A : the first is a heart. B : the second is a heart. C : the card is a king. Which pairs of these events are independent? Solution If A is true, then there is one fewer heart in the pack. The probability of B is 2. Hence 5 4 A and B are not independent. If A is true then C is neither more likely nor less likely than before. Its probability is still 3. Hence A and C are independent. Similarly the truth of B does not alter probability of C. The pairs A and C, B and C are independent 2. A women travels to work by bus, car or on foot with probabilities,, respectively. For each type of transport her probabilities of being late are,, arrives late one morning, find the probability that she come by bus. Solution respectively. If she
4 Here let L be the event that she is late, and B the event that she came by bus. Use the formula for conditional probability: P( B & L) PB L = PL ( ) P( B L) The probability that she came by bus is Exercises. Two cards are drawn without replacement from a pack. Events A, B,C,D are as follows : A : the first is an ace. B : the second is an 8. C : the first is red. D: the second is an spade Which pairs of events are independent? Which are exclusive? 2.Two dice are rolled. Events A, B, C, D are as follows: A : the first is an 5 B : the total is 8 C : the total is 7 D: the dice show the same number Which pairs of events are independent? Which are exclusive? 3. Two dice are rolled. Events A, B, C, D are as follows : A : the total is 7 B : the second die is a 2 C : both dice are less than 5 D: at least one die is a 6 Which pairs of events are independent? Which are exclusive? 4. Three fair coins are spun. Events A, B, C, D are as follows : A : the first coin is a head B : all the coins are heads C : there is at least one tail D: the first and last coins show the same Which pairs of events are independent? Which are exclusive? 5. With A, B, C, D as defined is Question, Find the following: (a) PB A (b) PD C 6. With A, B, C, D as defined is Question 2, Find the following: (a) PA B (b) PB D 7. With A, B, C, D as defined is Question 3, Find the following: (a) PA B (b) PC D 8. With A, B, C, D as defined is Question 4, Find the following: (a) PB A (b) PC A 9. A box contains 5 red and 6 blue marbles. Two are drawn without replacement. If the second is red find the probability that first was blue. 0. In his drawer a man has 7 left shoes and 0 right shoes. He picks two out at random. Find the probability that: (a) He has one left shoe and one right shoe (b) He has one left shoe and one right, given that the first was a left (c) The second is a left, given that he has one has one left and one right.
5 . A man travels to work by bus, car and motorcycle with probabilities 0.4, 0.5, 0. respectively. With each type of transport his chances of an accident are,, respectively. Find the probabilities that: (a) He goes by car and has an accident. (b) He does not have an accident. (c) He went by motorcycle, given that he had an accident. 2. % of the population has a certain disease. There is a test for the disease, which gives a positive response for 9 of the people with the disease, and for of the people without the 0 50 disease. A person is selected at random and tested. (a) What is the probability that the test gives a negative response? (b) If the test is positive, what is the probability that the person has the disease? (c) If the test is negative, what is the probability that the person does not have the disease? 3. An island contains two tribes; 2 3 of the population are Wache, who tell the truth with probability 0.7, and 3 are Oya, who tell the truth with probability 0.8. I meet a tribesman who tell me that he is a Wache. What is the probability that he is telling the truth? 4. In the certain town 3 4 of the voters are over 25, and they vote the Freedom Party with probability. Voters under 25 support the Freedom Party with probability 3. If a support of the Freedom Party is picked at random, what is the probability that he or she is under 25? 5. Events A and B are such that PA 0.4, PB 0.3, and Show that A and B are neither exclusive nor independent. Find PA B. 6. Events A and B are such that PA 0.3, PB 0.2, and PA or B 0.4 and B are not exclusive. Find PA & B and PA B. P A & B Show that A 7. Events A and B are such that PA 0.4, PB 0.3, and PA B 0.5. Find PA & B and PA or B Find PB A 8. Events A and B are such that PA 0.3, PB 0.5. Find PA or B and PA & B in the following cases: (a) A and B are exclusive (b) A and B are independent 20.3 Examination questions. A 2p coin a 0p are throw on a table. Event A is 'A head occurs on the 2p coin'. Event B is 'A head occurs on the 0p coin'. Event C is 'Two head or two tails obtained'. State, giving reasons, which of the following statements is (are) true and which is (are) false. (a) A and B are independent events (b) B and C are independent events. (c) A and C are mutually exclusive events
6 (d) A and BC are independent events. (e) PA B C = PA.P B.P C 2. Six balls colored yellow, green, brown, blue, pink, and black have values 2, 3, 4, 5, 6, 7 respectively. They are independent in size and placed in a box. Two balls are selected together from the box at random and the total number of point recorded (i) Find the probability that the total score is (a) 7, (b)9, (c)0, (d) greater 9, (e) odd. (ii) A game between two players, X and Y, starts with the six balls in a box. Each player in turn selects at random two balls, notes the score and then returns the balls to the box. The game is over when one of the players reaches a total score of 25 or more. (a) If X starts, calculate the probability that X wins on his second turn; (b) If Y starts, calculate the probability that Y wins on his second turn. [O ADD] 3. (a) The two electronic systems C,C2of a communications satellite operate independently and have probabilities of 0.and 0.05respectively of failing. Find the probability that (i) neither circuit fails (ii) at least one circuit fails, (iii) exactly one circuit fails (b) In a certain boxing competition all fights are either won or lost; draws are not permitted. If a boxer wins a fight then the probability that he wins his next fight is 3 ; if he loses 4 a fight the probability of him losing the next three fights is 2 3. Assuming that he won his last fight, use a tree diagram, or otherwise, to calculate the probability that of his three fights (i) he wins exactly two fights (ii) he wins at most two fights. State the most likely and least likely sequence of results for these three fights. 4. (i) The events A and B are such that PA 0.4, PB 0.45, P A B Show that the events A and B are neither mutually exclusive nor independent. (ii) A bag contains 2 red balls, 8 blue balls and 4 white balls. Three balls are taken from the bag at random and without replacement. Find the probability that all three balls are of the same colors. Find also the probability that all three balls are of different colors. 5. A box contains 25 apples, of which 20 are red and 5 are green. Of the red apples, 3 contains maggots and of the green apples, contains maggots. Two apples are chosen at random from the box. Find, in any order, (i) the probability that both apples contain maggots, (ii) the probability that both apples are red and at least one contains maggots, (iii) the probability that at least one apple contains maggots, given that both apples are red,
7 (iv) the probability that both apples are red given that at least one apple is red. 6. (a) Two digits X and Y are taken from a table of random sampling numbers. Event R is that X Y and events S is that X and Y are both less than 2. Write down (i) PR (ii) PRS (iii) PRS (iv) PR S (b) Conveyor belting for use in mines is tested for both strength and safety (the safety test is based on the amount of heat generated if the belt snaps). A testing station receives belting from three different suppliers : 30% of its tests are carried out on samples of belting from supplier A, 50% from B ; 20 % from C. From past experience the probability of failing the strength test is 0.02 for a sample from A, 0.2 from B and 0.04 from C. (i) What is the probability that a particular strength test will result in a failure? (ii) If a strength test result in a failure, What is the probability that the belting came from supplier A? (iii)what is the probability of a sample failing the safety tests given the following further information: supplier A  the probability of failing the safety tests is 0.05 and is independent of the probability of failing the strength test; supplier B % the probability of samples fail both strength and safety test supplier C exactly half the samples which fail the strength test also fail the safety test Common errors. Single probability If there are n outcomes to an experiment, then each has probability only if they are equally likely When two dice are rolled, there are possible for total score. But a total of 2 is less likely than total of 7,so neither has probability 2. Addition of probability The probability of 'A or 'B is only the sum of the probabilities if the events concerned are exclusive. In general: PA or B PA PB 3. Multiplication of probabilities The probability of A & B is only the product of the probabilities if the events concerned are independent. 4. Conditional probability independent. (a) Do not forgot to divide by PB when working out PA B (b) In the formula P A B do not assume that PA & Bis PA PB P( A& B) PB ( ) This is only true if A and B are
8 (c) Conditional probability is concerned with belief, not with cause and effect. If PA P A B. then it does not follow that B has caused A or prevented A. It may even be that B happened after A did. Solution (to exercise) (a) (e) 6 2. (a) 4 3. (a) (a) (a) (f) (b) 7 (b) 3 57 (b) (b) (b) 4 9 (c) (c) 9 7 (c) 6 (c) 5 6 (d) 5 2 (d) (d) (a) 9 (b) (a) 6 8. (a) (a) 32 (b) 5 (b) (b) 3 32 (b) 65 8 (b) (a) 8. (a) A& C, A& D, B & D indep. None exc. 2. A& C, A& D, B & D indep. B & C, C & D excl. 3. A& B indep. C& D exclusive. 4. A& Dindep. B& C excl. 5. (a) 4 (b) (c) (d) (a) 5 7. (a) 5 (b) 6 (b) 0
9 8. (a) 4 (b) (a) (b) 5 8 (c) 2. (a) (b) 5 8 (c) 2 2. (a) (b) (c) , ,0.55, (a) 0.8,0 (b) 0.65,0.5 =========================================================== References: Solomon, R.C. (997), A Level: Mathematics (4 th Edition), Great Britain, Hillman Printers(Frome) Ltd. More: (in Thai)
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