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Discrete Mathematics Exercises

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Problem 1 If 𝐴, 𝐵 and 𝐷 are three events such that 𝑃ሺ𝐴 ∪ 𝐵 ∪ 𝐷ሻ = 0.7, what is the value of 𝑃ሺ𝐴𝐶 ∩ 𝐵𝐶 ∩ 𝐷 𝐶 ሻ? 𝐴𝐶 = 𝑈 − 𝐴 Given we know { 𝐵𝐶 = 𝑈 − 𝐵 , 𝑤𝑒 𝑐𝑎𝑛 𝑑𝑟𝑎𝑤 𝑡ℎ𝑒 𝑑𝑖𝑎𝑔𝑟𝑎𝑚𝑠 𝑎𝑠 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔: 𝐷𝐶 = 𝑈 − 𝐷 a) 𝐴 ∪ 𝐵 ∪ 𝐷 b) 𝐴𝐶 ∩ 𝐵𝐶 ∩ 𝐷 𝐶 We can see 𝑃ሺ𝐴𝐶 ∩ 𝐵𝐶 ∩ 𝐷 𝐶 ሻ = 𝑃ሺ𝐴 ∪ 𝐵 ∪ 𝐷ሻ𝐶 Therefore, 𝑃ሺ𝐴𝐶 ∩ 𝐵𝐶 ∩ 𝐷 𝐶 ሻ = 1 − 0.7 = 0.3 Problem 2 Prove the following identity for any three events 𝐴, 𝐵, 𝑎𝑛𝑑 𝐶: 𝑃ሺ𝐴 ∪ 𝐵 ∪ 𝐶ሻ = 𝑃ሺ𝐴ሻ + 𝑃ሺ𝐵ሻ + 𝑃ሺ𝐶ሻ − 𝑃ሺ𝐴 ∩ 𝐵ሻ − 𝑃ሺ𝐴 ∩ 𝐶ሻ − 𝑃ሺ𝐵 ∩ 𝐶ሻ + 𝑃ሺ𝐴 ∩ 𝐵 ∩ 𝐶ሻ We can rewrite 𝑃ሺ𝐴 ∪ 𝐵 ∪ 𝐶ሻ 𝑎𝑠 𝑃(ሺ𝐴 ∪ 𝐵ሻ ∪ 𝐶). And we know for any two events 𝑋 𝑎𝑛𝑑 𝑌: 𝑃ሺ𝑋 ∪ 𝑌ሻ = 𝑃ሺ𝑋ሻ + 𝑃ሺ𝑌ሻ − 𝑃ሺ𝑋 ∩ 𝑌ሻ Using that, we can write: ...
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