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Chapter 5: Problem 50

Determine whether the following probabilities are computed using classical methods, empirical methods, or subjective methods. (a) The probability of having eight girls in an eight-child family is \(0.390625 \%\) (b) On the basis of a study of families with eight children, the probability of a family having eight girls is \(0.54 \%\) (c) According to a sports analyst, the probability that the Chicago Bears will win their next game is about \(30 \%\). (d) On the basis of clinical trials, the probability of efficacy of a new drug is \(75 \%\)

Short Answer

Expert verified
(a) Classical, (b) Empirical, (c) Subjective, (d) Empirical

Step by step solution

01

Identify Classical Probability Methods

Classical probability methods are used when each outcome in a sample space is equally likely. This often involves theoretical models or exact calculations based on mathematical principles.
02

Identify Empirical Probability Methods

Empirical probability methods are based on observed data or experimental results rather than pure theoretical calculations. These probabilities are derived from actual observations or experiments.
03

Identify Subjective Probability Methods

Subjective probability methods involve personal judgment, predictions, or estimates about how likely an event is to occur based on individual belief, intuition, or expertise.
04

Analyze Probability for Eight Girls in an Eight-Child Family

The probability of having eight girls in an eight-child family is provided as a percentage (0.390625%). This probability is computed using a mathematical model assuming each child has an equal chance of being a girl. Thus, it's a classical probability.
05

Analyze Probability Based on a Study of Families

The probability of a family with eight children having eight girls, given as 0.54%, is likely derived from observed data of such families. This makes it an empirical probability.
06

Analyze Sports Analyst's Prediction

The probability that the Chicago Bears will win their next game, given as 30%, is a prediction made by an analyst. This type of probability is subjective as it is based on the analyst's judgment and expertise.
07

Analyze Probability Based on Clinical Trials

The probability that a new drug's efficacy is 75% is based on clinical trial data. This makes it an empirical probability, as it is derived from experimental evidence.

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Key Concepts

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

Classical Probability
Classical probability is a method of calculating probabilities based on the assumption that all outcomes in a sample space are equally likely. This is often tied to theoretical models or purely mathematical calculations. Imagine flipping a fair coin. You have two possible outcomes: heads or tails, each with equal probability. Here are some other key points you should know:
  • It's often used in situations with clearly defined and simple outcomes.
  • It's the probability method used in games of chance, like rolling dice or drawing cards from a deck.
  • Formulas used: P(E) = Number of favorable outcomes / Total number of outcomes. For example, the likelihood of rolling a 3 on a standard six-sided die is 1/6.

In our given exercise, the probability of having eight girls in an eight-child family is 0.390625%. This uses a classical probability method because it leverages the mathematical assumption that each child has an equal chance (50%) of being a girl.
Empirical Probability
Empirical probability, also known as experimental probability, is grounded in real-world data collected from observations or experiments. This type of probability differs from classical probability because it's based on actual outcomes rather than theoretical assumptions.
  • It is often used in scientific research, such as clinical trials and behavioral studies.
  • Data is gathered from experiments, surveys, or historical records.
  • Formulas used: P(E) = Number of times event E occurs / Total number of trials. For instance, if a new drug is effective in 75 out of 100 patients, the empirical probability of the drug's efficacy is 75%.

In the exercise, the probability of a family with eight children having eight girls, given as 0.54%, is derived from actual data of families, making it an empirical probability. Similarly, the probability of a new drug's efficacy being 75% is based on clinical trials, which also falls under empirical probability.
Subjective Probability
Subjective probability is based on an individual's personal judgment, intuition, or expertise. Unlike classical and empirical probabilities, it does not rely on mathematical calculations or empirical data.
  • It involves predictions, beliefs, or best estimates about the likelihood of an event.
  • Often used in areas where statistical data is either unavailable or not applicable, such as market predictions or sports outcomes.
  • No specific formula exists; it is heavily reliant on personal or professional experience. For example, a financial expert estimating the probability of a stock price increase is using subjective probability.

In our exercise, the sports analyst's 30% prediction that the Chicago Bears will win their next game is a subjective probability. It is based on the analyst's knowledge, experience, and possibly other less tangible factors.

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Most popular questions from this chapter

In the Illinois Lottery game Little Lotto, an urn contains balls numbered 1 to \(30 .\) From this urn, 5 balls are chosen randomly, without replacement. For a \(\$ 1\) bet, a player chooses one set of five numbers. To win, all five numbers must match those chosen from the urn. The order in which the balls are selected does not matter. What is the probability of winning Little Lotto with one ticket?

Companies whose stocks are listed on the NASDAQ stock exchange have their company name represented by either four or five letters (repetition of letters is allowed). What is the maximum number of companies that can be listed on the NASDAQ?

Exclude leap years from the following calculations and assume each birthday is equally likely: (a) Determine the probability that a randomly selected person has a birthday on the 1 st day of a month. Interpret this probability. (b) Determine the probability that a randomly selected person has a birthday on the 31 st day of a month. Interpret this probability. (c) Determine the probability that a randomly selected person was born in December. Interpret this probability. (d) Determine the probability that a randomly selected person has a birthday on November \(8 .\) Interpret this probability. (e) If you just met somebody and she asked you to guess her birthday, are you likely to be correct? (f) Do you think it is appropriate to use the methods of classical probability to compute the probability that a person is born in December?

Among 21 - to 25 -year-olds, \(29 \%\) say they have driven while under the influence of alcohol. Suppose three \(21-\) to 25 -year-olds are selected at random. Source: U.S. Department of Health and Human Services, reported in USA Today (a) What is the probability that all three have driven while under the influence of alcohol? (b) What is the probability that at least one has not driven while under the influence of alcohol? (c) What is the probability that none of the three has driven while under the influence of alcohol? (d) What is the probability that at least one has driven while under the influence of alcohol?

The local golf store sells an "onion bag" that contains 35 "experienced" golf balls. Suppose the bag contains 20 Titleists, 8 Maxflis, and 7 Top-Flites. (a) What is the probability that two randomly selected golf balls are both Titleists? (b) What is the probability that the first ball selected is a Titleist and the second is a Maxfli? (c) What is the probability that the first ball selected is a Maxfli and the second is a Titleist? (d) What is the probability that one golf ball is a Titleist and the other is a Maxfli?
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