C. Symmetries

C. Symmetries - the drawing is without replacement . Then P...

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Handout C: Symmetries Ref: The Probability Tutoring Book , C. Ash, IEEE Press, NJ, 1993, pp. 17-18. Here are some typical symmetries: Draw cards either with or without replacement. P (ace on 1st draw) = P (ace 2nd)= P (ace 3rd), etc. P (ace 3rd, king 10th)= P (ace 1st, king 2nd)= P (ace 2nd, king 1st), etc. Intuitively, each position in the deck of cards has the same chance of harboring an ace; each pair of positions has the same chance of containing an ace and king, etc. Similarly, draw with or without replacement from a box containing red, while and black balls: P (RWBW drawn in that order)= P (WWRB)= P (WWRB)= P (RBWW), etc. Whatever the distribution of colors you draw, you are just as likely to get them in one order as another. It isn’t safe to rely on intuition, so I’ll do one proof as justification. I’ll show that P (ace on 3rd draw, king on 5th draw)= P (K on 1st, A on 2nd) This is immediate if the drawings are with replacemen t. Each probability is 4 52 ! 4 52 . Suppose
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Unformatted text preview: the drawing is without replacement . Then P (K on 1st, A on 2nd)= fav total = 4 52 ! 4 51 P (A on 3rd, K on 5th)= fav total = fav 52 ! 51 ! 50 ! 49 ! 48 For the fav, there are 5 slots to fill. The 3 rd slot can be filled in 4 ways, the 5 th slot in 4 ways, and then the other 3 slots in 50 ! 49 ! 48 ways. So P (A on 3rd, K on 5th)= 4 ! 4 ! 50 ! 49 ! 48 52 ! 51 ! 50 ! 49 ! 48 = 4 52 ! 4 51 The other 3 slots canceled out, leaving the same answer as P (K on 1st, A on 2nd) . Here is another way to solve this problem : P (K on 1st and A on 2nd)= P (K on 1st) P (A|K on 1st) where P (K on 1st)= 4 52 P (A|K on 1st)= 4 51 Example Draw without replacement from a box with 10 white and 5 black balls. To find the probability of W on the 1 st and 4 th draws (no information about the 2 nd and 3 rd ), take advantage of symmetry and switch to an easier problem: P (W on 1st and 4th)= P (W on 1st and 2nd)= 10 15 ! 9 14...
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C. Symmetries - the drawing is without replacement . Then P...

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