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Suppose you wanted to test your extrasensory perception (ESP) ability using an ordinary deck of 52 cards, which has 26 red and 26 black cards. You have a friend shuffle the deck and draw cards at random, replacing the card and reshuffling after each guess. You attempt to guess the color of each card. a. What is the probability that you guess the color correctly by chance? b. Is the answer in part (a) based on the relative-frequency interpretation of probability, or is it a personal probability? c. Suppose another friend has never tried the experiment but believes he has ESP and can guess correctly with probability. \(60 .\) Is the value of .60 a relativefrequency probability or a personal probability? Explain. d. Suppose another friend guessed the color of 1000 cards and got 600 correct. The friend claims she has ESP and has a .60 probability of guessing correctly. Is the value of .60 a relative-frequency probability or a personal probability? Explain.

Step by step solution

01

Determine the Probability of Guessing Correctly by Chance

Given an ordinary deck of 52 cards consisting of 26 red and 26 black cards, the probability of guessing the color of a card correctly is based on the possibility of it being either red or black. Since each card is equally likely to be red or black, the probability of guessing correctly by chance is:\[ P(\text{correct guess}) = \frac{1}{2} = 0.5 \]

02

Interpretation of the Probability

The probability calculated in the previous step (0.5) is a relative-frequency probability. This is because it is based on the frequency of red and black cards in repeated trials, following principles of classical probability as determined by the deck's distribution.

03

Analyze a Friend's Claim of ESP at 0.60 Probability

If another friend claims to have ESP and can guess correctly with a probability of 0.60 without prior experimentation, this value reflects a personal probability. It represents the friend's subjective belief in his ability rather than being based on empirical data or frequency.

04

Evaluate a Friend's Experimental Claim of 0.60 Probability

If another friend guesses the color correctly for 600 out of 1000 trials, this results in an experimental probability of:\[ P(\text{correct guess}) = \frac{600}{1000} = 0.6 \]This value, obtained from repeated trials, reflects a relative-frequency probability because it is based on empirical evidence from experimental data.

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Key Concepts

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

Relative-frequency Interpretation

The relative-frequency interpretation of probability is one way to understand how likely an event is to happen. It's all about observing outcomes over repeated trials or experiments. The more often you try something, like guessing the color of a card in a shuffled deck, the better idea you have of how often certain outcomes occur.
This interpretation assumes that you can estimate the probability of an event based on the frequency of its occurrence over a large number of trials.

• Imagine flipping a fair coin many times. If it lands on heads 500 times out of 1,000 flips, the relative-frequency probability of getting heads is 0.5.
• In our card guessing example, we calculated the probability of guessing correctly by chance as 0.5, based on the equal number of red and black cards in the deck.

In general, relative-frequency interpretation is grounded in actual experiments and observational data, rather than personal beliefs or speculative judgments.

Personal Probability

Personal probability is a bit different from relative-frequency probability. Here, probability is not based on empirical data, but rather on personal judgment or belief. This interpretation allows individuals to express how likely they think an event is, based on their own knowledge, experience, or intuition.
Suppose your friend believes they have extrasensory perception (ESP) and claims they can guess the color of a card with a 0.6 probability of being correct, without ever trying the guessing experiment. This belief is not yet supported by any experimental data.

• Personal probabilities can vary greatly from person to person because they are subjective estimates.
• They are influenced by a person's individual experiences, feelings, and confidence in their abilities.

While personal probabilities can guide personal decisions, they might not be reliable for making predictions about actual outcomes unless they are tested and confirmed through repeated experiments.

Classical Probability

Classical probability is one of the foundational approaches to understanding probability. It deals with situations where all outcomes are equally likely. This is great for simple scenarios where every outcome in a sample space has an equal chance of occurrence, like rolling a fair die or drawing a card from a well-shuffled deck.
With classic probability, calculations are straightforward because they rely on counting possible equally likely outcomes.

• For instance, the probability of picking a red card from a deck is determined by the ratio of red cards (26) to the total number of cards (52), yielding a probability of 0.5.
• Classical probability assumes a perfectly controlled environment where the conditions are consistent and unbiased.

This approach often supports and simplifies the reasoning in determining probabilities in games of chance or symmetrical scenarios. However, it might not always be applicable to real-world situations where conditions are more complex or outcomes are not equally likely.