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Here are the rules of NALGO: after the first card is placed on the table, each new card must have a higher number than the previous card. If it does, this new card remains on the table. If the new card is lower in value, then this card is removed from the table and the game is immediately over. Your final score is the number of cards that are placed on the table. For example, if n = 3, then there are six possible scenarios, each of them equally likely to occur. The cards placed on the table are marked in bold. First Card Second Card Third Card 1 2 3 1 3 2 1 3 1 2 2 3 3 2 3 1 2 1 Cards on Table 1, 2, 3 1, 3 2 2, 3 3 3 Points 3 2 1 2 1 1 Thus, if there are n = 3 cards in the deck, then you score 1 point with probability, 2 points with probability, and the maximum 3 points with probability. And so, the expected value of your final score is 1 x + 2x +3× = ¹0 = 1.666. (a) If there are n = 4 cards in the deck, determine the expected value of your final score. Show your calculations. (b) Prove that for all positive integers n ≥3, if there are n cards in the deck, you score exactly 1 point with probability and exactly 2 points with probability. (c) Prove that for all positive integers n and k with n>k, if there are n cards in the deck, you score exactly k points with probability(+1! (Hint: convince yourself that (k+1)! = (+1)!). (d) If there are infinitely many cards in the deck (i.e., n→ ∞o), determine the expected value of your final score. Hint: you will find it helpful to use the identity e² = 1 + + + + + +=+ Recall that e = 2.71828..., the base of the natural logarithm. ....

Here are the rules of NALGO: after the first card is placed on the table, each new card must have a higher number than the previous card. If it does, this new card remains on the table. If the new card is lower in value, then this card is removed from the table and the game is immediately over. Your final score is the number of cards that are placed on the table. For example, if n = 3, then there are six possible scenarios, each of them equally likely to occur. The cards placed on the table are marked in bold. First Card Second Card Third Card 1 2 3 1 3 2 1 3 1 2 2 3 3 2 3 1 2 1 Cards on Table 1, 2, 3 1, 3 2 2, 3 3 3 Points 3 2 1 2 1 1 Thus, if there are n = 3 cards in the deck, then you score 1 point with probability, 2 points with probability, and the maximum 3 points with probability. And so, the expected value of your final score is 1 x + 2x +3× = ¹0 = 1.666. (a) If there are n = 4 cards in the deck, determine the expected value of your final score. Show your calculations. (b) Prove that for all positive integers n ≥3, if there are n cards in the deck, you score exactly 1 point with probability and exactly 2 points with probability. (c) Prove that for all positive integers n and k with n>k, if there are n cards in the deck, you score exactly k points with probability(+1! (Hint: convince yourself that (k+1)! = (+1)!). (d) If there are infinitely many cards in the deck (i.e., n→ ∞o), determine the expected value of your final score. Hint: you will find it helpful to use the identity e² = 1 + + + + + +=+ Recall that e = 2.71828..., the base of the natural logarithm. ....

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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You are given a deck of n cards, which are numbered 1 through n. After the n cards are randomly shuffled, the cards are dealt face up on the table, one card at a time. Here are the rules of NALGO: after the first card is placed on the table, each new card must have a higher number than the previous card. If it does, this new card remains on the table. If the new card is lower in value, then this card is removed from the table and the game is immediately over. Your final score is the number of cards that are placed on the table. For example, if n = 3, then there are six possible scenarios, each of them equally likely to occur. The cards placed on the table are marked in bold. First Card Second Card Third Card 1 2 3 1 3 2 2 2 3 3 1 3 1 2 3 1 2 1 Cards on Table 1, 2, 3 1, 3 2 2, 3 3 3 Points 3 2 1 2 1 1 Thus, if there are n = 3 cards in the deck, then you score 1 point with probability , 2 points with probability, and the maximum 3 points with probability. And so, the expected value of your final score is 1 x + 2x +3× == 1.666. (a) If there are n = 4 cards in the deck, determine the expected value of your final score. Show your calculations. (b) Prove that for all positive integers n ≥ 3, if there are n cards in the deck, you score exactly 1 point with probability and exactly 2 points with probability. (c) Prove that for all positive integers n and k with n > k, if there are n cards in the deck, you score exactly k points with probability (+1)! (Hint: convince yourself that (k+1)! = (k+1)!). (d) If there are infinitely many cards in the deck (i.e., n→ ∞), determine the expected value of your final score. Hint: you will find it helpful to use the identity e² = 1 + + + + + Recall that e = 2.71828...., the base of the natural logarithm. +=+
Transcribed Image Text:You are given a deck of n cards, which are numbered 1 through n. After the n cards are randomly shuffled, the cards are dealt face up on the table, one card at a time. Here are the rules of NALGO: after the first card is placed on the table, each new card must have a higher number than the previous card. If it does, this new card remains on the table. If the new card is lower in value, then this card is removed from the table and the game is immediately over. Your final score is the number of cards that are placed on the table. For example, if n = 3, then there are six possible scenarios, each of them equally likely to occur. The cards placed on the table are marked in bold. First Card Second Card Third Card 1 2 3 1 3 2 2 2 3 3 1 3 1 2 3 1 2 1 Cards on Table 1, 2, 3 1, 3 2 2, 3 3 3 Points 3 2 1 2 1 1 Thus, if there are n = 3 cards in the deck, then you score 1 point with probability , 2 points with probability, and the maximum 3 points with probability. And so, the expected value of your final score is 1 x + 2x +3× == 1.666. (a) If there are n = 4 cards in the deck, determine the expected value of your final score. Show your calculations. (b) Prove that for all positive integers n ≥ 3, if there are n cards in the deck, you score exactly 1 point with probability and exactly 2 points with probability. (c) Prove that for all positive integers n and k with n > k, if there are n cards in the deck, you score exactly k points with probability (+1)! (Hint: convince yourself that (k+1)! = (k+1)!). (d) If there are infinitely many cards in the deck (i.e., n→ ∞), determine the expected value of your final score. Hint: you will find it helpful to use the identity e² = 1 + + + + + Recall that e = 2.71828...., the base of the natural logarithm. +=+
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