13.7 Conditional probability\nThe probability that an event occurs given that another event has already occurred is called conditional probability.\n\nThe probability that event B occurs, given that event A has already occurred is denoted by P(B | A). The symbol Ã¢â‚¬Ëœ|Ã¢â‚¬â„¢ stands for Ã¢â‚¬ËœgivenÃ¢â‚¬â„¢.\n\nThe formula for conditional probability is:\n\n\nConditional probability: where the probability of an event is conditional (depends) on another event occurring first. For two events A and B, the conditional probability of event B, given that event A occurs, is denoted by P(B| A) and can be calculated using the formula: \nÃ¢â‚¬â€°\nÃ¢â‚¬â€°\n\nWORKED EXAMPLE 20\nThis Venn diagram below shows the results of a survey where students were asked to indicate whether they liked apples or bananas.\n\nimages\nIf one student is selected at random:\n\nWhat is the probability that the student likes bananas?\n\nWhat is the probability that the student likes bananas, given that they also like apples?\n\nComment on any differences between the answers for parts a and b.\n\nTHINK	WRITE\na	1	Find the total number of students.	a	Total number of students in survey\nÃ¢â‚¬Æ’= 12 + 7 + 10 + 4\nÃ¢â‚¬Æ’= 33\n2	Find the total number of students who like bananas.		Total number who like bananas\nÃ¢â‚¬Æ’= 7 + 10Ã¢â‚¬Æ’\nÃ¢â‚¬Æ’= 17\n3	Determine the probability using the correct formula.		P(bananas) = P(B)\n \n4	Write the answer.		The probability that a student likes bananas is .\nb	1	Determine the number of students who like apples.	b	Number of students who like apples\nÃ¢â‚¬Æ’= 12 + 7\nÃ¢â‚¬Æ’= 19\n2	Find the probability that a student likes apples.		P(apples) = P(A)\n \n4953	Note the number liking both apples and bananas. This is the overlapping region of the two sets.		Number who like both apples and bananas\nÃ¢â‚¬Æ’= n(A Ã¢Ë†Â© B)\nÃ¢â‚¬Æ’= 7\n4	Determine the probability a student likes both apples and bananas.		\n5	Apply the formula to determine the conditional probability.		\n6	Write the answer.		The probability that a student likes bananas, given that they also like apples is .\nThis answer is also supported by the figures in the Venn diagram.\nc	Why arenÃ¢â‚¬â„¢t the answers for parts a and b both the same?	c	The answer for part a determines the proportion of students who like bananas out of the whole group of students. The part b answer gives the proportion of students who like bananas out of those who like apples.\nNote: These probabilities could also be expressed as decimals or percentages.\n\nIt is possible to transpose the conditional probability formula to determine P(A Ã¢Ë†Â© B).\n\nP(A Ã¢Ë†Â© B) = P(A) Ãƒâ€” P(B | A)\n\nWORKED EXAMPLE 21\nIn a student survey, the probability that a student likes apples is . The probability that a student likes bananas, given that they also like apples, is . What is the probability that a student selected at random likes both apples and bananas?\n\nTHINK	WRITE\n1	Write the given information.	\n2	Apply the rearranged conditional probability formula.	\n3	Answer the question.	The probability that a student selected at random likes both apples and bananas is .\n496Conditional probability can also be determined by examining outcomes from a tree diagram.\n\nWORKED EXAMPLE 22\nThree coins are flipped simultaneously.\n\nDisplay the outcomes as a tree diagram.\n\nDetermine the probability that a Head will result from the third coin, given that the first two coins resulted in a Head (H) and a Tail (T).\n\nTHINK	WRITE/DRAW\na	Draw a tree diagram to display the flipping of three coins. Write the individual outcomes.	a	images\nb	1	Look for the outcomes where the first two flips resulted in a Head and a Tail.	b	There are four outcomes where the first two flips are a Head and a Tail Ã¢â‚¬â€ HTH, HTT, THH and THT.\n2	How many of these outcomes have a Head for the third flip?		There are two of these outcomes where the third flip resulted in a Head Ã¢â‚¬â€ HTH and THH.\n3	Calculate the probability.		From four possible outcomes, two satisfy the conditions.\nP(H on third flip | H and T on first two flips)\nA two-way table can also be used to determine conditional probability.\n\nWORKED EXAMPLE 23\nTwo dice are rolled and the numbers are added.\n\nShow the results in a two-way table.\n\nDetermine the probability that the sum of the two dice is 7, given that their total is greater than 6.\n\n497THINK	WRITE/DISPLAY\na	Show the results of rolling two dice in a two-way table.	a	\nDie 2\n1	2	3	4	5	6\nDie 1	1	(1, 1)	(1, 2)	(1, 3)	(1, 4)	(1, 5)	(1, 6)\n2	(2, 1)	(2, 2)	(2, 3)	(2, 4)	(2, 5)	(2, 6)\n3	(3, 1)	(3, 2)	(3, 3)	(3, 4)	(3, 5)	(3, 6)\n4	(4, 1)	(4, 2)	(4, 3)	(4, 4)	(4, 5)	(4, 6)\n5	(5, 1)	(5, 2)	(5, 3)	(5, 4)	(5, 5)	(5, 6)\n6	(6, 1)	(6, 2)	(6, 3)	(6, 4)	(6, 5)	(6, 6)\nb	1	Which outcomes have a total greater than 6?	b	There are 36 outcomes. 21 of these have a total greater than 6 (6, 1), (5, 2), (4, 3), (3, 4), (2, 5), (1, 6), (6, 2) . . . etc.\n2	Which of these outcomes have a total equal to 7?		There are 6 of these outcomes that have a total of 7 Ã¢Ë†â€™ (6, 1), (5, 2), (4, 3), (3, 4), (2, 5), (1, 6).\n3	Write the probability.		\nExercise 13.7 Conditional probability\nicon\nINDIVIDUAL PATHWAYS\n   PRACTISE\nQuestions:\n1Ã¢â‚¬â€œ3, 5, 7, 9Ã¢â‚¬â€œ11	   CONSOLIDATE\nQuestions:\n1Ã¢â‚¬â€œ4, 6Ã¢â‚¬â€œ13	   MASTER\nQuestions:\n1Ã¢â‚¬â€œ15\n         Individual pathway interactivity int-4539 icon\nREFLECTION\n\nHow can you determine when a probability question is a conditional one?\n\nFLUENCY\nWE21 A group of motocross racers was asked to comment on which of two tracks, A or B, they used. The results were recorded in the Venn diagram below.\n\nimages\nHow many motocross racers were surveyed?\n\nCalculate P(A Ã¢Ë†Â© B).\n\nCalculate:\n\nP(A)\n\nP(B | A).\n\nCalculate:\n\nP(B)\n\nP(A | B).\n\n498Consider your answers to question 1.\n\nUse your answers from part c to determine P(A Ã¢Ë†Â© B).\n\nUse your answers from part d to determine P(A Ã¢Ë†Â© B).\n\nComment on your answers to parts a and b in this question.\n\nimages\nA survey was conducted to determine whether a group of students preferred drink A or drink B. The results of the survey produced the following probabilities.\n\n and . Determine P(A Ã¢Ë†Â© B).\n\nWE22 Two fair coins are tossed.\n\nDisplay the outcomes as a tree diagram.\n\nDetermine the probability that a Head results on the second coin, given that the first coin also resulted in a Head.\n\nWE23 Two standard dice are rolled and the numbers are added together.\n\nShow the results in a two-way table.\n\nDetermine the probability that the sum of the two dice is even, given that their total is greater than 7.\n\nUNDERSTANDING\nimages\nA group of 40 people was surveyed regarding the types of movies, comedy or drama, that they enjoyed. The results are shown below.\n\n28 enjoyed comedy.\n\n17 enjoyed drama.\n\n11 liked both comedy and drama.\n\n6 did not like either type.\n\nDraw a Venn diagram to display the results of the survey.\n\nDetermine the probability that a person selected at random:\n\nlikes comedy\n\nlikes drama\n\nlikes both comedy and drama\n\nlikes drama, given that they also like comedy\n\nlikes comedy, given that they also like drama.\n\nArrange the probabilities in part b in order from least probable to most probable.\n\nA teacher gave her class two tests. Only 25% of the class passed both tests, but 40% of the class passed the first test. What percentage of those who passed the first test also passed the second test?\n\nREASONING\nIf P(A) = 0.3, P(B) = 0.5 and P(A Ã¢Ë†Âª B) = 0.6, calculate:\n\nP(A Ã¢Ë†Â© B)\n\nP(B | A)\n\nP(A | B).\n\nA group of 80 boys is auditioning for the school musical. They are all able to either sing, play a musical instrument, or both. Of the group, 54 can play a musical instrument and 35 are singers. What is the chance that if a randomly selected student is a singer he can also play a musical instrument?\n\n499A white die and a black die are rolled. The dice are 6-sided and unbiased. Consider the following events.\n\nEvent A: the white die shows a 6.\n\nEvent B: the black die shows a 2.\n\nEvent C: the sum of the two dice is 4.\n\nDetermine the following probabilities.\n\nP(A | B)\n\nP(B | A)\n\nP(C | A)\n\nP(C | B)\n\nA die is rolled and the probability of rolling a 6 is . However, with the condition that the number