TREE DIAGRAMS AND LAW OF TOTAL PROBABILITY

1 TREE DIAGRAMS AND LAW OF TOTAL PROBABILITYiii. Given t...
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1 TREE DIAGRAMS AND LAW OF TOTAL PROBABILITYiii. Given that an order does not have a mistake, find the probability that it was Anu who handled it.

2 A 0.4 B 0.35 0.25 C

3 0.05 𝑀|𝐴 𝐴 0.95 0.4 𝑀 β€² |𝐴 𝐡 0.35 0.25 𝐢

4 𝐴 𝐡 𝐢 𝑀|𝐴 0.05 0.95 𝑀 β€² |𝐴 𝑀|𝐡 𝑀 β€² |𝐡 0.25 𝑀|𝐢 0.97 𝑀′|𝐢 0.4 0.04 0.350.96 𝑀 β€² |𝐡 0.25 0.03 𝑀|𝐢 𝐢 0.97 𝑀′|𝐢

5 =𝟎.πŸŽπŸ’πŸπŸ“ Pr 𝑀 = Pr 𝐴 Γ— Pr 𝑀 𝐴 + Pr 𝐡 Γ— Pr 𝐡 𝑀 +Pr⁑(𝐢)Γ—Pr⁑(𝐢|𝑀) ii.=𝟎.πŸ’Γ—πŸŽ.πŸŽπŸ“+𝟎.πŸ‘πŸ“Γ—πŸŽ.πŸŽπŸ’+𝟎.πŸπŸ“Γ—πŸŽ.πŸŽπŸ‘ =𝟎.πŸŽπŸ’πŸπŸ“

6 iii. Given that an order does not have a mistake, find the probability that it was Anu who handled it.

7 iii. Given that an order does not have a mistake, find the probability that it was Anu who handled it. CONDITIONAL PROBABILITY Pr 𝐴 𝑀 β€² = Pr⁑(𝐴∩ 𝑀 β€² ) Pr⁑( 𝑀 β€² ) Pr 𝐴 𝑀 β€² = Pr⁑(𝐴)Γ—Pr⁑(𝑀′|𝐴) Pr⁑(𝑀′) Pr 𝐴 𝑀 β€² = 0.4Γ—0.95 1βˆ’ β‰ˆ0.3965

8 INDEPENDENT EVENTS ii. Pr⁑(𝐡|𝐴) iii. Pr⁑(𝐴∩𝐡)If A and B are independent events: Can A and B both occur at the same time? If Pr 𝐴 =0.6, Pr 𝐡 =0.8, and A and B are both independent, state the values of: i. Pr 𝐴 𝐡 ii. Pr⁑(𝐡|𝐴) iii. Pr⁑(𝐴∩𝐡)

9 INDEPENDENT EVENTS ii. Pr⁑(𝐡|𝐴) iii. Pr⁑(𝐴∩𝐡)If A and B are independent events: Can A and B both occur? Yes. Since they are independent, they do not affect each other and so if one occurs it does not influence the likelihood that the other will also occur. If Pr 𝐴 =0.6, Pr 𝐡 =0.8, and A and B are both independent, state the values of: i. Pr 𝐴 𝐡 ii. Pr⁑(𝐡|𝐴) iii. Pr⁑(𝐴∩𝐡) π‘Ίπ’Šπ’π’„π’† 𝑨 𝒂𝒏𝒅 𝑩 𝒂𝒓𝒆 π’Šπ’π’…π’†π’‘π’†π’π’…π’†π’π’•, 𝐏𝐫 𝑨 𝑩 = 𝐏𝐫 𝑨 =𝟎.πŸ” π‘Ίπ’Šπ’π’„π’† 𝑨 𝒂𝒏𝒅 𝑩 𝒂𝒓𝒆 π’Šπ’π’…π’†π’‘π’†π’π’…π’†π’π’•, 𝐏𝐫 𝑩 𝑨 = 𝐏𝐫 𝑩 =𝟎.πŸ– π‘Ίπ’Šπ’π’„π’† 𝑨 𝒂𝒏𝒅 𝑩 𝒂𝒓𝒆 π’Šπ’π’…π’†π’‘π’†π’π’…π’†π’π’•, 𝐏𝐫 π‘¨βˆ©π‘© =𝐏 𝐫 𝑨 ×𝐏 𝐫 𝑩 =𝟎.πŸ”Γ—πŸŽ.πŸ–=𝟎.πŸ’πŸ–

10 MUTUALLY EXCLUSIVE EVENTSIf C and D are mutually exclusive events: Can C and D both occur at the same time? b. If Pr 𝐢 =0.27 and Pr 𝐡 =0.1 and C and D are mutually exclusive events, determine the following probabilities: i. Pr 𝐢 𝐷 ii. Pr⁑(𝐷|𝐢) iii. Pr⁑(𝐢∩𝐷)

11 MUTUALLY EXCLUSIVE EVENTSIf C and D are mutually exclusive events: Can C and D both occur at the same time? b. If Pr 𝐢 =0.27 and Pr 𝐡 =0.1 and C and D are mutually exclusive events, determine the following probabilities: i. Pr 𝐢 𝐷 ii. Pr⁑(𝐷|𝐢) iii. Pr⁑(𝐢∩𝐷) = 0 NO!! Mutually exclusive means that if one occurs, the other is automatically impossible. 𝐏𝐫 π‘ͺ 𝑫 =𝟎 C D 𝐏𝐫 𝑫 π‘ͺ =𝟎

12 THE ADDITION LAW The Addition Law is a faithful friend because, unlike the law of Independent events, the Addition Law is ALWAYS true, whether or not the events are independent. Given that 𝐏𝐫 𝑨 =𝟎.πŸ•, 𝐏𝐫 𝑩 =𝟎.πŸ“ 𝐚𝐧𝐝 𝐏𝐫 π‘¨βˆ©π‘© =𝟎.πŸ‘: Are events A and B independent? Calculate the value of 𝐏𝐫 𝑨βˆͺ𝑩

13 𝐏𝐫 𝑨βˆͺ𝑩 =𝟎.πŸ•+𝟎.πŸ“βˆ’πŸŽ.πŸ‘=𝟎.πŸ— THE ADDITION LAWThe Addition Law is a faithful friend because, unlike the law of Independent events, the Addition Law is ALWAYS true, whether or not the events are independent. Given that 𝐏𝐫 𝑨 =𝟎.πŸ•, 𝐏𝐫 𝑩 =𝟎.πŸ“ 𝐚𝐧𝐝 𝐏𝐫 π‘¨βˆ©π‘© =𝟎.πŸ‘: Are events A and B independent? No, they are not. 𝐏𝐫 𝑨 ×𝐏𝐫 𝑩 =𝟎.πŸ•Γ—πŸŽ.πŸ“=𝟎.πŸ‘πŸ“ and 𝐏𝐫 π‘¨βˆ©π‘© =𝟎.πŸ‘ Since 𝐏𝐫(π‘¨βˆ©π‘©)≠𝐏𝐫(𝑨)×𝐏𝐫(𝑩), they are NOT independent. Calculate the value of 𝐏𝐫 𝑨βˆͺ𝑩 𝐏𝐫 𝑨βˆͺ𝑩 =𝐏𝐫 𝑨 +𝐏𝐫 𝑩 βˆ’ππ«(π‘¨βˆ©π‘©) 𝐏𝐫 𝑨βˆͺ𝑩 =𝟎.πŸ•+𝟎.πŸ“βˆ’πŸŽ.πŸ‘=𝟎.πŸ—

14 KARNAUGH MAPS and VENN DIAGRAMSSolve the following question from a CAS Free exam:

15 KARNAUGH MAPS and VENN DIAGRAMSSolve the following question from a CAS Free exam: B Bο‚’ A 1 8 1 5 Aο‚’ ? 4 5 1 3 2 3 1 𝐏𝐫 𝑨 β€² βˆ©π‘© = 𝟏 πŸ‘ βˆ’ 𝟏 πŸ– = πŸ“ πŸπŸ’ A B When A and B are mutually exclusive events, π΄β€²βˆ©π΅ is simply equal to 𝐡. So, 𝐏𝐫 𝑨 β€² βˆ©π‘© = 𝟏 πŸ‘ 1 5 1 3