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Chapter 5: Problem 31

You read in a book about bridge that the probability that each of the four players is dealt exactly one ace is about 0.11. This means that (a) in every 100 bridge deals, each player has one ace exactly 11 times. (b) in one million bridge deals, the number of deals on which each player has one ace will be exactly 110,000. (c) in a very large number of bridge deals, the percent of deals on which each player has one ace will be very close to 11%.(d) in a very large number of bridge deals, the average number of aces in a hand will be very close to 0.11. (e) None of these

Short Answer

Expert verified
The answer is (c). In a very large number of deals, 11% will have each player with one ace.

Step by step solution

01

Understanding Probability

The problem states that the probability of each player getting exactly one ace in a bridge hand is about 0.11. This means that if you deal a very large number of hands, the proportion of hands where each player receives one ace will be close to 0.11.
02

Interpreting Probability in 100 Deals

Let's evaluate option (a). If the probability is 0.11, it suggests that in a set of 100 bridge deals, you expect 11 of those to result in each player having one ace. This is an expectation, not certainty.
03

Interpreting Probability in 1,000,000 Deals

Now let's evaluate option (b). With the probability of 0.11, out of one million deals, you would expect 110,000 deals to result in each player having one ace. Again, this is based on probability not guaranteed occurrences.
04

Interpreting Long-term Probability

Evaluate option (c). In a very large number of bridge deals, the law of large numbers suggests that the percentage of deals where each player has one ace will approximate 11%. This is a correct interpretation of probability.
05

Average Aces in a Hand Interpretation

Evaluate option (d). An average number of 0.11 aces per hand means that if each hand contributes 1 ace distributed throughout hands, it is not the same as each hand having exactly one ace; thus, this interpretation is incorrect.
06

Eliminating Incorrect Options

Options (a), (b), and (d) are incorrect interpretations of the probability. (a) assumes certainty in a finite sample, (b) projects exact occurrences from probability, and (d) misinterprets average occurrence with specific hand composition.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interpretation of Probability
Probability is the measure of the likelihood that an event will occur. It's quantified as a number between 0 and 1, where 0 indicates impossibility, and 1 indicates certainty. In the context of the bridge problem, a probability of 0.11 indicates that, in many repeats of the experiment, each player can expect to have exactly one ace in about 11% of the games.
  • This doesn't guarantee outcomes but provides an average expectation over many trials.
  • The essence of probability is uncertainty management, where an event with a probability of 0.11 means it's likely to occur in roughly 11 out of 100 dealings if repeated enough times.
  • It's crucial to understand that probability is not about exact outcomes in short runs; it's about expected frequency in large numbers of trials.
When dealing with real-life applications, recognizing the probabilistic nature of events helps us anticipate results over time, rather than focusing on specific short-run occurrences.
Law of Large Numbers
The Law of Large Numbers (LLN) is a fundamental concept in probability and statistics. It states that as a sample size grows, its mean gets closer to the average of the whole population. For example:
  • In our bridge deals context, the law implies that as you deal more hands, the percentage of hands where each player gets one ace will approach 11%.
  • This is because random variations balance out over a large number of observations or trials.
  • The law highlights the stability of frequencies over time, providing assurance of the averaged outcomes predicted by probability when the sample size is large.
The LLN helps us understand why interpreting a probability of 0.11 is more accurate over a vast number of games rather than a small sample, thereby remedying misinterpretations of probability that expect specific results even in smaller samples.
Bridge Deals
Bridge is a card game typically involving four players and a standard 52-card deck. During a game:
  • Cards are dealt evenly, which means each player gets 13 cards.
  • The probability challenge discussed involves each player receiving exactly one ace in their 13 cards.
  • This scenario emphasizes the randomness and distribution factors inherent in card games.
Understanding bridge deals in terms of probability involves recognition of the distribution of cards and the combinatorial challenges present. Since there are 4 aces in a deck, the chance that each player ends up with precisely one ace hinges on a very specific set of conditions being met, making its direct occurrence relatively rare but predictably frequent across many repeated deals.
Average Expectation
Average expectation refers to the mean outcome one can anticipate over many trials, considering random variables. In probabilistic terms for bridge:
  • If you state that the probability of each player having one ace is 0.11, it serves as an estimate of the long-term proportion of such deals.
  • This doesn't mean every player gets an ace in every eleventh deal; rather, over a sufficiently large number of games, 11% will have this outcome.
  • Misinterpreting expectation can lead one to incorrectly assume exact and consistent results within a small number of trials, which detracts from understanding true probability dynamics.
Hence, average expectation in probability is crucial: it tempers predictions to align with what's statistically realistic over a broad spectrum of events, rather than in isolated cases.

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