3 nusfosdsap lecture 12 12 gem2900 understanding

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Unformatted text preview: at is the probability that at least one couple gets to pair together? Let Mi = ith couple gets to pair together = P(M1) + P(M 2 ) + P(M 3 ) − P(M1 ∩ M 2 ) − P(M1 ∩ M 3 ) − P(M 2 ∩ M 3 ) + P(M1 ∩ M 2 ∩ M 3 ) P(M1 ∪ M 2 ∪ M 3 ) = 3× 1 ⎛ 3⎞ 1 1 + − ⎜ ⎟× 3 ⎜ 2⎟ 3× 2 3× 2×1 ⎝⎠ = 1− 11 + 2! 3! NUS/FOS/DSAP Lecture 12 – 12 GEM2900 - Understanding Uncertainty & Statistical Thinking Semester 2, 2010/2011 How to Calculate with Probabilities How if 4 couples attend a dance class were paired off at random, what is the probability that at least one couple gets to pair together? Let Mi be the event that ith couple gets to pair together P(M1 ∪ M 2 ∪ M 3 ∪ M 4 ) = Matching Problem ∑ P(M i i ) − ∑ P(M i ∩ M j ) + i< j i< j< k ∑ P(M i ∩ M j ∩ Mk) − P(M1 ∩ M 2 ∩ M 3 ∩ M 4 ) ⎛ 4⎞ 1 ⎛ 4⎞ 1 1 1 − + ⎜ ⎟× = 4× − ⎜ ⎟× 4 ⎜ 2⎟ 4× 3 ⎜ 3⎟ 4× 3× 2 4× 3× 2×1 ⎝⎠ ⎝⎠ = 1− 111 +− 2! 3! 4! NUS/FOS/DSAP Lecture 12 – 13 GEM2900 - Understanding Uncertainty & Statistical Thinking Semester 2, 2010/2011 How to Calculate with Probabilities How if 5 or 8 couples attend a dance class were paired off at random, what is the probability that at least one couple gets to pair together? Let Mi be the event that ith couple gets to pair together Matching Problem 1111 P(M1 ∪ M 2 ∪ M 3 ∪ M 4 ∪ M 5 ) = 1 − + − + ≈ 0 .6333 2! 3! 4! 5! P(M1 ∪ M 2 ∪ M 3 ∪ M 4 ∪ M 5 ∪ M 6 ∪ M 7 ∪ M 8 ) 1111111 ≈ 0 .6321 = 1− + − + − + − 2! 3! 4! 5! 6! 7! 8! NUS/FOS/DSAP Lecture 12 – 14 GEM2900 - Understanding Uncertainty & Statistical Thinking Semester 2, 2010/2011 How to Calculate with Probabilities With n = 5 couples, the probability Matching Problem of ‘at least one match’ is about 0.63, and, after that, the probability of ‘at least one match’ stays very close to 0.63 as n increases. Q. What is the probability that at least one couple gets to dance together if there are n couples attending a dance class and everyone is paired of at random? 1− 111 1 + − + − ... ± 2! 3! 4! n! Note: When n ≥ 7, we have an accuracy up to decimal place of ten thousandths (0.6321) NUS/FOS/DSAP Lecture 12 – 15...
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