Questions: In a certain card game, the probability that a player is dealt a particular hand is 0.46 . Explain what this probability means. If you play this card game 100 times, will you be dealt this hand exactly 46 times? Why or why not? Choose the correct answer below. A. The probability 0.46 means that exactly 46 out of every 100 dealt hands will be that particular hand. Yes, you will be dealt this hand exactly 46 times since the probability refers to long-term behavior, not short-lerm. B. The probability 0.46 means that approximately 46 out of every 100 dealt hands will be that particular hand. No, you will not be dealt this hand exactly 46 times since the probability refers to what is expected in the long-term, not short-term. C. The probability 0.46 means that exactly 46 out of every 100 dealt hands will be that particular hand. Yes, you will be dealt this hand exactly 46 times since the probability refers to short-term behavior, not long-term. D. The probability 0.46 means that approximatoly 46 out of every 100 dealt hands will be that particular hand. No, you will not be deatt this hand exactly 46 times since the probability reters to what is expected in the short-term, not long-term.

In a certain card game, the probability that a player is dealt a particular hand is 0.46 . Explain what this probability means. If you play this card game 100 times, will you be dealt this hand exactly 46 times? Why or why not? Choose the correct answer below. A. The probability 0.46 means that exactly 46 out of every 100 dealt hands will be that particular hand. Yes, you will be dealt this hand exactly 46 times since the probability refers to long-term behavior, not short-lerm. B. The probability 0.46 means that approximately 46 out of every 100 dealt hands will be that particular hand. No, you will not be dealt this hand exactly 46 times since the probability refers to what is expected in the long-term, not short-term. C. The probability 0.46 means that exactly 46 out of every 100 dealt hands will be that particular hand. Yes, you will be dealt this hand exactly 46 times since the probability refers to short-term behavior, not long-term. D. The probability 0.46 means that approximatoly 46 out of every 100 dealt hands will be that particular hand. No, you will not be deatt this hand exactly 46 times since the probability reters to what is expected in the short-term, not long-term.
Transcript text: In a certain card game, the probability that a player is dealt a particular hand is 0.46 . Explain what this probability means. If you play this card game 100 times, will you be dealt this hand exactly 46 times? Why or why not? Choose the correct answer below. A. The probability 0.46 means that exactly 46 out of every 100 dealt hands will be that particular hand. Yes, you will be dealt this hand exactly 46 times since the probability refers to long-term behavior, not short-lerm. B. The probability 0.46 means that approximately 46 out of every 100 dealt hands will be that particular hand. No, you will not be dealt this hand exactly 46 times since the probability refers to what is expected in the long-term, not short-term. C. The probability 0.46 means that exactly 46 out of every 100 dealt hands will be that particular hand. Yes, you will be dealt this hand exactly 46 times since the probability refers to short-term behavior, not long-term. D. The probability 0.46 means that approximatoly 46 out of every 100 dealt hands will be that particular hand. No, you will not be deatt this hand exactly 46 times since the probability reters to what is expected in the short-term, not long-term.
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Solution

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Solution Steps

Step 1: Calculate the Expected Number of Occurrences

To calculate the expected number of occurrences of the event, we use the formula: \[E = p \times n\] Where \(p = 0.46\) is the probability of the event occurring in a single trial, and \(n = 100\) is the number of trials. Substituting the given values, we get: \[E = 0.46 \times 100 = 46\]

Step 2: Understanding the Law of Large Numbers

The Law of Large Numbers in probability theory states that as the number of trials \(n\) increases, the relative frequency of an event's occurrence gets closer to the theoretical probability \(p\) of that event. However, this law does not guarantee that the event will occur exactly \(p \times n\) times in every set of \(n\) trials. Instead, it provides a long-term average expectation. The actual number of occurrences is subject to variability and may not exactly match the expected number \(E\), especially in smaller sample sizes.

Final Answer:

While the expected number of occurrences of the event is \(46\), the actual number can vary due to the randomness inherent in probabilistic events. This variability is a natural part of probabilistic systems, and the Law of Large Numbers suggests that the relative frequency of occurrence will get closer to \(p\) as \(n\) becomes very large.

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