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A gambler began playing blackjack with $110 in chips. After exactly 12 hands, he left the table with $320 in chips, aving won some hands and lost others. Each win earned $100 and each loss cost $10. How many possible utcomes were there for the first 5 hands he played? (For example, won the first hand, lost the second, etc.)
- a)10
- b)18
- c)26
- d)32
- e)64
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
A gambler began playing blackjack with $110 in chips. After exactly 12...
Let W be the number of wins and L be the number of losses. Since the total number of hands equals 12 and the net winnings equal $210, we can construct and solve the following simultaneous equations: w + l = 12, 100 w – 10 l = 210. so l = 9, w = 3. So we know that the gambler won 3 hands and lost 9. We do not know where in the sequence of 12 hands the 3 wins appear. So when counting the possible outcomes for the first 5 hands, we must consider these possible scenarios: 1) Three wins and two losses 2) Two wins and three losses 3) One win and four losses 4) No wins and five losses In the first scenario, we have WWWLL. We need to know in how many different ways we can arrange these five letters: 5!/2!3! = 10. So there are 10 possible arrangements of 3 wins and 2 losses. The second scenario --WWLLL -- will yield the same result: 10. The third scenario -- WLLLL -- will yield 5 possible arrangements, since the one win has only 5 possible positions in the sequence. The fourth scenario --LLLLL -- will yield only 1 possible arrangement, since rearranging these letters always yields the same sequence. Altogether, then, there are 10 + 10 + 5 + 1 = 26 possible outcomes for the gambler's first five hands.
Most Upvoted Answer
A gambler began playing blackjack with $110 in chips. After exactly 12...
Given Information:
The gambler started with $110 in chips and ended with $320 in chips after 12 hands of blackjack. Each win earned $100 and each loss cost $10.
Objective:
To find the number of possible outcomes for the first 5 hands the gambler played.
Approach:
We can solve this problem using the concept of combinations. Since the gambler played 12 hands of blackjack, the number of wins and losses can vary. However, the total number of wins and losses must add up to 12.
Solution:
To determine the number of possible outcomes for the first 5 hands, we need to find the number of ways we can distribute the wins and losses among these 5 hands.
Let's consider the number of wins in the first 5 hands. It can range from 0 to 5, as the gambler can win any number of hands from 0 to 5. The remaining hands will be losses.
We can calculate the number of possible outcomes by finding the number of ways to choose the number of wins in the first 5 hands and then distributing the wins and losses among these hands.
Calculating the number of possible outcomes:
- Number of wins in the first 5 hands: 0, 1, 2, 3, 4, 5
- Number of losses in the first 5 hands: 5, 4, 3, 2, 1, 0
We can use the combination formula to calculate the number of ways to choose the wins:
Number of ways to choose 'r' wins out of 'n' hands = nCr = n! / (r! * (n-r)!)
For each possible number of wins in the first 5 hands, we can calculate the number of ways to distribute the wins and losses among these hands:
- For 0 wins in the first 5 hands, there is 1 way to distribute the losses (5 losses).
- For 1 win in the first 5 hands, there are 5 ways to distribute the win (choose the position of the win) and 1 way to distribute the losses (4 losses).
- For 2 wins in the first 5 hands, there are 10 ways to distribute the wins and 1 way to distribute the losses (3 losses).
- For 3 wins in the first 5 hands, there are 10 ways to distribute the wins and 1 way to distribute the losses (2 losses).
- For 4 wins in the first 5 hands, there are 5 ways to distribute the wins and 1 way to distribute the losses (1 loss).
- For 5 wins in the first 5 hands, there is 1 way to distribute the wins and 1 way to distribute the losses (0 losses).
Total number of possible outcomes:
To calculate the total number of possible outcomes, we sum up the number of ways for each possible number of wins in the first 5 hands.
Total number of possible outcomes = 1 + 5 + 10 + 10 + 5 + 1 = 32
Therefore, there are 32 possible outcomes for the first 5 hands the gambler played.
Conclusion:
The correct answer is option (d) 32
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A gambler began playing blackjack with $110 in chips. After exactly 12 hands, he left the table with $320 in chips,aving won some hands and lost others. Each win earned $100 and each loss cost $10. How many possible utcomes were there for the first 5 hands he played? (For example, won the first hand, lost the second, etc.)a)10b)18c)26d)32e)64Correct answer is option 'C'. Can you explain this answer?
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A gambler began playing blackjack with $110 in chips. After exactly 12 hands, he left the table with $320 in chips,aving won some hands and lost others. Each win earned $100 and each loss cost $10. How many possible utcomes were there for the first 5 hands he played? (For example, won the first hand, lost the second, etc.)a)10b)18c)26d)32e)64Correct answer is option 'C'. Can you explain this answer? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A gambler began playing blackjack with $110 in chips. After exactly 12 hands, he left the table with $320 in chips,aving won some hands and lost others. Each win earned $100 and each loss cost $10. How many possible utcomes were there for the first 5 hands he played? (For example, won the first hand, lost the second, etc.)a)10b)18c)26d)32e)64Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A gambler began playing blackjack with $110 in chips. After exactly 12 hands, he left the table with $320 in chips,aving won some hands and lost others. Each win earned $100 and each loss cost $10. How many possible utcomes were there for the first 5 hands he played? (For example, won the first hand, lost the second, etc.)a)10b)18c)26d)32e)64Correct answer is option 'C'. Can you explain this answer?.
A gambler began playing blackjack with $110 in chips. After exactly 12 hands, he left the table with $320 in chips,aving won some hands and lost others. Each win earned $100 and each loss cost $10. How many possible utcomes were there for the first 5 hands he played? (For example, won the first hand, lost the second, etc.)a)10b)18c)26d)32e)64Correct answer is option 'C'. Can you explain this answer? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A gambler began playing blackjack with $110 in chips. After exactly 12 hands, he left the table with $320 in chips,aving won some hands and lost others. Each win earned $100 and each loss cost $10. How many possible utcomes were there for the first 5 hands he played? (For example, won the first hand, lost the second, etc.)a)10b)18c)26d)32e)64Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A gambler began playing blackjack with $110 in chips. After exactly 12 hands, he left the table with $320 in chips,aving won some hands and lost others. Each win earned $100 and each loss cost $10. How many possible utcomes were there for the first 5 hands he played? (For example, won the first hand, lost the second, etc.)a)10b)18c)26d)32e)64Correct answer is option 'C'. Can you explain this answer?.
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A gambler began playing blackjack with $110 in chips. After exactly 12 hands, he left the table with $320 in chips,aving won some hands and lost others. Each win earned $100 and each loss cost $10. How many possible utcomes were there for the first 5 hands he played? (For example, won the first hand, lost the second, etc.)a)10b)18c)26d)32e)64Correct answer is option 'C'. Can you explain this answer?, a detailed solution for A gambler began playing blackjack with $110 in chips. After exactly 12 hands, he left the table with $320 in chips,aving won some hands and lost others. Each win earned $100 and each loss cost $10. How many possible utcomes were there for the first 5 hands he played? (For example, won the first hand, lost the second, etc.)a)10b)18c)26d)32e)64Correct answer is option 'C'. Can you explain this answer? has been provided alongside types of A gambler began playing blackjack with $110 in chips. After exactly 12 hands, he left the table with $320 in chips,aving won some hands and lost others. Each win earned $100 and each loss cost $10. How many possible utcomes were there for the first 5 hands he played? (For example, won the first hand, lost the second, etc.)a)10b)18c)26d)32e)64Correct answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
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