solution03 - Solutions to Homework 03 [1] The number of...

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Unformatted text preview: Solutions to Homework 03 [1] The number of ways of picking 20 raccoons out of N is N 20 ! The number of ways of picking 5 tagged raccoons out of 10 and 15 untagged raccoons out of N-10 is 10 5 ! N- 10 15 ! The probability of picking 5 tagged out of 20 is p ( N ) = 10 5 ! N- 10 15 ! N 20 ! To find the N that maximizes p ( N ), we need to find the range of N where p ( N ) increases with N. This is equivalent to the range of N where p ( N ) /p ( N- 1) ≥ 1. p ( N ) p ( N- 1) = N- 10 15 ! N- 1 20 ! N 20 ! N- 11 15 ! = ( N- 10)( N- 20) ( N- 25) N ≥ 1 ( N- 10)( N- 20) ≥ ( N- 25) N ⇒ 40 ≥ N Therefore, when N = 40, p ( N ) has the maximum of p (40) = . 284. [2] (i) P [ A | B ] = P [ A ∩ B ] P [ B ] for P [ B ] > 0. Now ≤ P [ A ∩ B ] , < P [ B ] ⇒ ≤ P [ A | B ] Also, ( A ∩ B ) ⊂ B ⇒ P [ A ∩ B ] ≤ P [ B ] ⇒ P [ A | B ] ≤ 1 Therefore, ≤ P [ A | B ] ≤ 1 ii. First note that B ∩ S = B . Then P [ S | B ] = P [ B ∩ S ] P [ B ] = P [ B ] P [ B ] = 1 1 iii. We’ll use the following result...
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This note was uploaded on 09/03/2008 for the course ECSE 4510 taught by Professor Woods during the Spring '03 term at Rensselaer Polytechnic Institute.

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solution03 - Solutions to Homework 03 [1] The number of...

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