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4.4 Conditional Probability When additional information is known that will affect the probability of an outcome, we calculate what is called a conditional probability. π(π΄ β© π΅) π(π΄|π΅) = π(π΅) π(π΄|π΅) is read βthe probability of event A, given that the event B has occurredβ. Once B is known, then the event set for A is restricted to those outcomes in the set (π΄ β© π΅). So the outcome set is reduced from S to B. Example 1 What is the probability of rolling a sum greater than 7 with two dice if it is known that the first die rolled is a 3? π(π π’π > 7|1π π‘ ππππ ππ 3) = 2β 36 1 6 2 6 = × 36 1 12 = 36 1 = 3 π(π π’π > 7 β© 1π π‘ ππππ ππ 3) π(1π π‘ ππππ ππ 3) = Example 2 If a family is chosen at random from the set of all families with exactly two children, determine the probability that a) The family has two boys given that at least one child is a boy π(2 πππ¦π |πππ ππ π πππ¦) 1st 2nd π(2 πππ¦π β© πππ ππ π πππ¦) boy BB = boy π(πππ ππ π πππ¦) girl BG 4 outcomes 1β 4 π(π) = 4 = 3β boy GB 4 girl 1 4 1 = × = GG girl 4 3 3 b) The family has two boys if it is known that the first child is a boy π(2 πππ¦π |1π π‘ ππ π πππ¦) π(2πππ¦π β© 1π π‘ ππ π πππ¦) 1β4 = = 2β π(1π π‘ ππ π πππ¦) 4 1 2 =4×1=½ Since π(π΅) × π(π΄|π΅) = π(π΄β©π΅) π(π΅) × π(π΅), then π(π΅) × π(π΄|π΅) = π(π΄ β© π΅) Multiplication Law for Conditional Probability π(π΄ β© π΅) = π(π΄|π΅) × π(π΅) Example 3 If the first card drawn from a standard deck of cards is not replaced, what is the probability ofβ¦ a) Drawing an ace on the first draw AND a jack on the second draw? π(2ππ ππππ β© 1π π‘ πππ) = π(2ππ ππππ|1π π‘ πππ) × π(1π π‘ πππ) 4 4 = × 51 52 16 4 = = 2652 663 b) Drawing a pair of eights? π(2ππ 8 β© 1π π‘ 8) = π(2ππ 8|1π π‘ 8) × π(1π π‘ 8) 3 4 12 1 = × = = 51 52 2652 221 Homework p. 235 # 1, 2, 4, 6, 7, 9, 10