1. Suppose P ( A ) = 0.45, P ( B ) = 0.40, P ( C ) = 0.50, P ( A ∩ B ) = 0.15, P ( A ∩ C ) = 0.20, P ( B ∩ C ) = 0.25, P ( A ∩ B ∩ C ) = 0.05. a) Find P ( A ∪ B ∪ C ). b) Find P ( A ∪ ( B ∩ C ) ). c) Find P ( A ∩ ( B ∪ C ) ). 2. Find the value of p that would make this a valid probability model. a) Suppose S = { 0, 2, 4, 6, 8, … } ( even non-negative integers ) and P ( 0 ) = p, P ( k ) = k 31, k = 2, 4, 6, 8, … . b) Suppose S = { 1, 2, 3, 4, … } ( positive integers ) and P ( 1 ) = p, P ( k ) = ( )! 3lnkk, k = 2, 3, 4, … . 3. Suppose S = { 3, 4, 5, 6, … } and P ( k ) = kC 5, k = 3, 4, 5, 6, … . a) Find the value of C that would make this a valid probability model. b) Find P ( even outcome ).4. At Initech, 60% of all employees surf the Internet during work hours. 24% of the employees surf the Internet and play MMORPG during work hours. It is also known that 72% of the employees either surf the Internet or play MMORPG (or both) during work hours. a) What proportion of the employees play MMORPG during work hours? b) If it is known that an employee surfs the Internet during work hours, what is the probability that he/she also plays MMORPG ? c) Suppose an employee does not play MMORPG during work hours. What is the probability that he/she surfs the Internet?1. Suppose P ( A ) = 0.45, P ( B ) = 0.40, P ( C ) = 0.50, P ( A ∩ B ) = 0.15, P ( A ∩ C ) = 0.20, P ( B ∩ C ) = 0.25, P ( A ∩ B ∩ C ) = 0.05. a) Find P ( A ∪ B ∪ C ). P ( A ∪ B ∪ C ) = 0.80. OR P ( A ∪ B ∪ C ) = P ( A ) + P ( B ) + P ( C ) – P ( A ∩ B ) – P ( A ∩ C ) – P ( B ∩ C ) + P ( A ∩ B ∩ C ) = 0.45 + 0.40 + 0.50 – 0.15 – 0.20 – 0.25 + 0.05 = 0.80. b) Find P ( A ∪ ( B ∩ C ) ). P ( A ∪ ( B ∩ C ) ) = 0.65.OR P ( A ∪ ( B ∩ C ) ) = P ( A ) + P ( B ∩ C ) – P ( A ∩ ( B ∩ C ) ) = = 0.45 + 0.25 – 0.05 = 0.65. c) Find P ( A ∩ ( B ∪ C ) ). P ( A ∩ ( B ∪ C ) ) = 0.30. 2. Find the value of p that would make this a valid probability model. a) Suppose S = { 0, 2, 4, 6, 8, … } ( even non-negative integers ) and P ( 0 ) = p, P ( k ) = k 31, k = 2, 4, 6, 8, … . 1 = p ( 0 ) + p ( 2 ) + p ( 4 ) + p ( 6 ) + p ( 8 ) + … = p + 8642 31313131+++ + … = p + 91191− = p + 81. ⇒ p = 87.b) Suppose S = { 1, 2, 3, 4, … } ( positive integers ) and P ( 1 ) = p, P ( k ) = ( )! 3lnkk, k = 2, 3, 4, … . Must have ()∑xxp all = 1. ⇒ p ( 1 ) + ( )∑∞=2! 3lnkkk = 1. ( )∑∞=2! 3lnkkk = ( )∑∞=0! 3lnkkk – 1 – ln 3 = e ln 3 – 1 – ln 3 = 2 – ln 3. p ( 1 ) + 2 – ln 3 = 1. ⇒ p ( 1 ) = ln 3 – 1 ≈ 0.0986. 3. Suppose S = { 3, 4, 5, 6, … } and P ( k ) = kC 5, k = 3, 4, 5, 6, … . a) Find the value of C that would make this a valid probability model. ∑∞=3 5 kkC = basetermfirst−1 = 511 53 −C = 54 125 C = 100C = 1. ⇒ C = 100. b) Find P ( even outcome ). P ( even outcome ) = p ( 4 ) + p ( 6 ) + p ( 8 ) + … = ...510051005100864 +++ = basetermfirst−1 = 511 510024 − = 24 4 = 61.4. At Initech, 60% of all employees surf the Internet during work hours. 24% of the employees surf the Internet and play MMORPG during work hours. It is also known that 72% of the employees either surf the Internet or play MMORPG (or both) during work hours. P( Internet ) = 0.60, P( Internet ∩ MMORPG ) = 0.24, P( Internet ∪ MMORPG ) = 0.72. a) What proportion of the employees play MMORPG during work hours? P( Internet ∪ MMORPG ) = P( Internet ) + P( MMORPG ) – P( Internet ∩ MMORPG ) 0.72 = 0.60 + P( MMORPG ) – 0.24 P( MMORPG ) = 0.36. MMORPG MMORPG' Internet 0.24 0.36 0.60 Internet' 0.12 0.28 0.40 0.36 0.64 1.00 b) If it is known that an employee surfs the Internet during work hours, what is the probability that he/she also plays MMORPG ? P( MMORPG | Internet ) = 60.024.0)Internet P()Internet MMORPG P( =∩ = 0.40. c) Suppose an employee does not play MMORPG during work hours. What is the probability that he/she surfs the Internet? 169===∩640360) MMORPG P() MMORPG Internet P() MMORPG Internet P(|..''' =
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