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.Card rule: 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.***The image is the example***Questions:Prove that for all positive integers n ≥ 3, if there are n cards in the deck, you score exactly 1 point with probability 12 and exactly 2 points with probability 13. For example, if n = 3, then there are six possible scenarios, each of them equally likely to occur. Thecards placed on the table are marked in bold.First CardSecond Card Third CardCards on Table Points1, 2, 31, 313121321212, 323131331Thus, if there are n =probability3 cards in the deck, then you score 1 point with probability , 2 points wfinaland the maximum 3 points with probability . And so, the expected value ofyourScore is 1 X+2 x+3 x =10||1.666.

Need to prove that for all positive integers n ≥ 3, if there are n cards in the deck, you score…

Answered: You are given a deck of n cards, which…

College Algebra

10th Edition

ISBN:9781337282291

Author:Ron Larson

Publisher:Ron Larson

Chapter8: Sequences, Series,and Probability

Section8.7: Probability

Problem 6ECP: In Pennsylvania’s Cash 5 game, a player chooses five different numbers from 1 to 43. If these five...

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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.

Card rule: 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.

***The image is the example***

Questions:

Prove that for all positive integers n ≥ 3, if there are n cards in the deck, you score exactly 1 point with probability 12 and exactly 2 points with probability 13.

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
Cards on Table Points
1, 2, 3
1, 3
1
3
1
2
1
3
2
1
2
1
2, 3
2
3
1
3
1
3
3
1
Thus, if there are n =
probability
3 cards in the deck, then you score 1 point with probability , 2 points w
final
and the maximum 3 points with probability . And so, the expected value of
your
Score is 1 X
+2 x+3 x =
10
||
1.666.

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