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**The Law of Total Probability **

Let A1, A2 and A3 be three mutually exclusive and exhaustive events and let B denote some other event. Note for B to occur it has to be in conjunction with at least one of A1, A2 and A3. Therefore, by applying the multiplication rule to a general situation of this type we obtain the following. The Law of Total Probability Let A1, A2, ,An be a set of mutually exclusive and exhaustive events. If B is any other event it follows that 𝑃 𝐵 =𝑃 𝐵 𝐴 1 +𝑃 𝐵 𝐴 P 𝐵 𝐴 𝑗 𝑗=1 𝑛 𝑃 𝐵 𝐴 𝑗 =

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**The Law of Total Probability **

Drawing a Venn Diagram of the situation can often help with the understanding of the problem. For example, three mutually exclusive and exhaustive events A1, A2, A3 and any other event B could be represented by the following Venn diagram 𝐴 1 B 𝐴 2 𝐴 3

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**The Law of Total Probability **

Let A1, A2 and A3 be three mutually exclusive and exhaustive events and let B denote some other event. Note for B to occur it has to be in conjunction with at least one of A1, A2 and A3. Therefore, by applying the multiplication rule to a general situation of this type we obtain the following. The Law of Total Probability Let A1, A2, ,An be a set of mutually exclusive and exhaustive events. If B is any other event it follows that 𝑃 𝐵 =𝑃 𝐵 𝐴 1 +𝑃 𝐵 𝐴 P 𝐵 𝐴 𝑗 𝑗=1 𝑛 𝑃 𝐵 𝐴 𝑗 =

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**The Law of Total Probability **

Example 1 There are three boxes, each containing a different number of light bulbs. The first box has 10 bulbs, of which four are dead, the second has six bulbs, of which one is dead, and the third box has eight bulbs of which three are dead. What is the probability of a dead bulb being selected when a bulb is chosen at random from one of the three boxes? 𝑃 𝐷 = 𝐃 Box 1 4 10 D 1 3 = 1 3 𝐃 Box 2 1 6 D 1 3 𝐃 Box 3 3 8 D

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**The Law of Total Probability **

Let A1, A2 and A3 be three mutually exclusive and exhaustive events and let B denote some other event. Note for B to occur it has to be in conjunction with at least one of A1, A2 and A3. Therefore, by applying the multiplication rule to a general situation of this type we obtain the following. The Law of Total Probability Let A1, A2, ,An be a set of mutually exclusive and exhaustive events. If B is any other event it follows that 𝑃 𝐵 =𝑃 𝐵 𝐴 1 +𝑃 𝐵 𝐴 P 𝐵 𝐴 𝑗 𝑗=1 𝑛 𝑃 𝐵 𝐴 𝑗 =

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**The Law of Total Probability **

Example 2 Three boxes contain red and green balls. Box 1 has 5 red balls* and 5 green balls*, Box 2 has 7 red balls* and 3 green balls* and Box 3 contains 6 red balls* and 4 green balls*. The respective probabilities of choosing a box are 1/4, 1/6, 1/8. What is the probability that the ball chosen is green? 5 10 3 10 4 10

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**The Law of Total Probability **

Solution We begin by defining the following sets. Let, G = the ball chosen is green. B1 = Box 1 is selected B2 = Box 2 B3 = Box 3 Then P(G|B1) = 5/10, P(G|B2) = 3/10 and P(G|B3) = 4/10. Therefore, using the law of total probability we have 5 10 1 4 3 10 1 6 4 10 1 8 𝑃 𝑮 = × × × 1 8 = 9 40

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**The Law of Total Probability **

Let A1, A2 and A3 be three mutually exclusive and exhaustive events and let B denote some other event. Note for B to occur it has to be in conjunction with at least one of A1, A2 and A3. Therefore, by applying the multiplication rule to a general situation of this type we obtain the following. The Law of Total Probability Let A1, A2, ,An be a set of mutually exclusive and exhaustive events. If B is any other event it follows that 𝑃 𝐵 =𝑃 𝐵 𝐴 1 +𝑃 𝐵 𝐴 P 𝐵 𝐴 𝑗 𝑗=1 𝑛 𝑃 𝐵 𝐴 𝑗 =

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**The Law of Total Probability **

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