Chapter 4probability And Probability Modelscopyright 2018 Mcgraw Hill ✓ Solved

Chapter 4 Probability and Probability Models 1 Chapter Outline 4.1 Probability, Sample Spaces, and Probability Models 4.2 Probability and Events 4.3 Some Elementary Probability Rules 4.4 Conditional Probability and Independence 4.5 Bayes’ Theorem (Optional) 4.6 Counting Rules (Optional) .1 Probability, Sample Spaces, and Probability Models An experiment is any process of observation with an uncertain outcome The possible outcomes for an experiment are called the experimental outcomes Probability is a measure of the chance that an experimental outcome will occur when an experiment is carried out The sample space of an experiment is the set of all possible experimental outcomes The experimental outcomes in the sample space are called sample space outcomes LO4-1: Define a probability, a sample space, and a probability model.

Probability If E is an experimental outcome, then P(E) denotes the probability that E will occur and: Conditions 0 ï‚£ P(E) ï‚£ 1 such that: If E can never occur, then P(E) = 0 If E is certain to occur, then P(E) = 1 The probabilities of all the experimental outcomes must sum to 1 LO Assigning Probabilities to Sample Space Outcomes Classical method For equally likely outcomes Relative frequency method Using the long run relative frequency Subjective method Assessment based on experience, expertise or intuition LO LO4-1 Probability Models Probability model: a mathematical representation of a random phenomenon Random variable: a variable whose value is numeric and is determined by the outcome of an experiment Probability distribution: A probability model describing a random variable Discrete probability distributions (Chapter 6) Continuous probability distributions (Chapter LO4-1 Some Important Probability Distributions Discrete probability distributions Binomial distribution Poisson distribution Continuous probability distributions Normal distribution Exponential distribution .2 Probability and Events An event is a set of sample space outcomes The probability of an event is the sum of the probabilities of the sample space outcomes If all outcomes equally likely, the probability of an event is just the ratio of the number of outcomes that correspond to the event divided by the total number of outcomes LO4-2: List the outcomes in a sample space and use the list to compute probabilities.

LO4-2 Classical Method Example 4.1 A newly married couple plans to have two children Would like to know all possible outcomes BB BG GB GG Want to know probabilities Assuming all equal P(BB) = P(BG) = P(GB) = P(GG) = ¼ 4-9 LO4-2 Subjective Example 4.3 A company is choosing a new CEO There are four candidates Adams (A) Chung (C) Hill (H) Rankin (R) An industry analysts feels the probabilities are: P(A) = 0.1 P(C) = 0.2 P(H) = 0.5 P(R) = 0..3 Some Elementary Probability Rules Complement Union Intersection Addition Conditional probability Multiplication 4-11 LO4-3: Use elementary profitability rules to compute probabilities. 11 LO4-3 Complement Figure 4.3 The complement () of an event A is the set of all sample space outcomes not in A P() = 1 – P(A) Union and Intersection The union of A and B are elementary events that belong to either A or B or both Written as A  B The intersection of A and B are elementary events that belong to both A and B Written as A ∩ B LO LO4-3 Some Elementary Probability Rules Figure 4.

LO4-3 Mutually Exclusive Figure 4.5 A and B are mutually exclusive if they have no sample space outcomes in common In other words: P(A ∩ B) = The Addition Rule If A and B are mutually exclusive, then the probability that A or B (the union of A and B) will occur is P(A  B) = P(A) + P(B) If A and B are not mutually exclusive: P(A  B) = P(A) + P(B) – P(A ∩ B) where P(A ∩ B) is the joint probability of A and B both occurring together LO.4 Conditional Probability and Independence The probability of an event A, given that the event B has occurred, is called the conditional probability of A given B Denoted as P(A|B) Further, P(A|B) = P(A ∩ B) / P(B) P(B) ≠0 LO4-4: Compute conditional probabilities and assess independence.

The General Multiplication Rule There are two ways to calculate P(A ∩ B) Given any two events A and B P(A ∩ B) = P(A) P(B|A) and P(A ∩ B) = P(B) P(A|B) LO Interpretation Restrict sample space to just event B The conditional probability P(A|B) is the chance of event A occurring in this new sample space In other words, if B occurred, then what is the chance of A occurring LO Independence of Events Two events A and B are said to be independent if and only if: P(A|B) = P(A) This is equivalent to P(B|A) = P(B) Assumes P(A) and P(B) greater than zero LO The Multiplication Rule The joint probability that A and B (the intersection of A and B) will occur is P(A ∩ B) = P(A) • P(B|A) = P(B) • P(A|B) If A and B are independent, then the probability that A and B will occur is: P(A ∩ B) = P(A) • P(B) = P(B) • P(A) LO LO4-4 Contingency Tables Table 4..5 Bayes’ Theorem (Optional) S1, S2, …, Sk represents k mutually exclusive possible states of nature, one of which must be true P(S1), P(S2), …, P(Sk) represents the prior probabilities of the k possible states of nature If E is a particular outcome of an experiment designed to determine which is the true state of nature, then the posterior (or revised) probability of a state Si, given the experimental outcome E, is calculated using the formula on the next slide LO4-5: Use Bayes’ Theorem to update prior probabilities to posterior probabilities (Optional).

Bayes’ Theorem Continued LO LO4-5 Example 4.14 The Oil Drilling Case: Site Selection Example 4.14 Oil company trying to decide about drilling There are three states of nature No oil (S1) P(S1) = 0.7 Some oil (S2) P(S2) = 0.2 Much oil (S3) P(S3) = 0.1 Company can perform a seismic experiment Gives three readings, low, medium, and high P(high|none) = 0.04 P(high|some) = 0.02 P(high|much) = 0.96 Assume site gives high reading Wish to revise the prior probabilities LO4-5 Example 4.14 The Oil Drilling Case: Site Selection Continued Example 4..6 Counting Rules (Optional) A counting rule for multiple-step experiments (n1)(n2)…(nk) A counting rule for combinations LO4-6: Use some elementary counting rules to compute probabilities (Optional).

LO4-6 A Tree Diagram of Answering Three True–False Questions Figure 4. ))P(E|S+P(S...)+)P(E|S)+P(S)P(E|SP(S ))P(E|SP(S P(E) ))P(E|SP(S P(E) E)P(S =|E)P(S kk ii iii i 2211    75000 . . . . 0 ) ( ) | ( ) ( ) ( ) ( ) | ( 03125 . . . . 0 ) ( ) | ( ) ( ) ( ) ( ) | ( 21875 . . . . 0 ) ( ) | ( ) ( ) ( ) ( ) | ( 128 . . . . . . . 0 ( ) | ( ) | ( ) ( ) | ( ) ( ) ( ) ( ) ( ) ( = = = à‡ = = = = à‡ = = = = à‡ = = + + = + + = à‡ + à‡ + à‡ = high P much high P much P high P high much P high much P high P some high P some P high P high some P high some P high P none high P none P high P high none P high none P much high P some high P some P none high P none P high much P high some P high none P high P

Paper for above instructions

Understanding Probability and Probability Models

Probability is a pivotal concept in statistics that provides a mathematical framework for quantifying uncertainty in various situations, ranging from simple games of chance to complex business decisions. This paper explores the fundamental notions of probability, sample spaces, events, elementary probability rules, and the relationships between events, alongside the application of Bayes' Theorem.

1. Definitions and Concepts

Probability refers to a measure that quantifies the likelihood of an event occurring within a defined sample space. Mathematically, for an event E, the probability P(E) lies within the range of 0 to 1:
- P(E) = 0 indicates that the event cannot occur,
- P(E) = 1 signifies that the event is certain to happen,
- The total probabilities of all outcomes in the sample space must sum to 1 (McClave & Sincich, 2018).
A sample space is defined as the collection of all possible outcomes from an experiment. For instance, when tossing a coin, the sample space S can be described as S = {Heads, Tails}. If we consider an experiment of rolling a six-sided die, the sample space is S = {1, 2, 3, 4, 5, 6} (Mendenhall, Beaver, & Beaver, 2019).
A probability model integrates both the sample space and the probabilities of each outcome. For example, in a fair die roll, the probability of each outcome can be assigned as P(1) = P(2) = … = P(6) = 1/6, which perfectly illustrates a uniform probability model (Blitzstein & Hwang, 2015).

2. Assigning Probabilities

There are various methods to assign probabilities:
- Classical Method involves recognizing equally likely outcomes.
- Relative Frequency Method estimates probabilities based on historical data and the frequency of events over time.
- Subjective Method is based on personal judgment or experience, which can lead to probabilistic assessments when numerical data is scarce (Tversky & Kahneman, 1974).

3. Events and Probabilities

An event is defined as a subset of outcomes from a sample space. Suppose we have the experiment of rolling two coins. The sample space includes:
- HH (both heads)
- HT (first head, second tail)
- TH (first tail, second head)
- TT (both tails)
If we define event A as obtaining at least one head, A = {HH, HT, TH}. The probability of event A can be calculated as:
\[ P(A) = P(HH) + P(HT) + P(TH) = \frac{1}{4} + \frac{1}{4} + \frac{1}{4} = \frac{3}{4} \]
This computation aligns with the notion that if all outcomes are equally likely, the probability of an event is simply the ratio of favorable outcomes to total outcomes (DeGroot & Schervish, 2012).

4. Elementary Probability Rules

Complement Rule

The complement of an event A, denoted by A', consists of all outcomes not in A. The probability of the complement can be calculated using:
\[ P(A') = 1 - P(A) \]

Union and Intersection

- Union (A ∪ B) signifies occurrences of either event A or event B or both. The probability of the union of two events is given by:
\[ P(A \cup B) = P(A) + P(B) - P(A ∩ B) \]
- Intersection (A ∩ B) reflects events happening simultaneously. The probability of joint occurrence aids in understanding dependencies between events (Schafer, 2015).

5. Conditional Probability and Independence

Conditional Probability refers to the probability of an event A given that another event B has occurred:
\[ P(A|B) = \frac{P(A ∩ B)}{P(B)} \]
This helps refine predictions based on previous occurrences. For example, knowing the weather conditions can alter the probability of attending an outdoor event (Hogg & Tanis, 2001).
Independence of Events is established if the occurrence of one event does not influence the other. Mathematically, A and B are independent if:
\[ P(A|B) = P(A) \quad \text{or} \quad P(B|A) = P(B) \]
This implies:
\[ P(A ∩ B) = P(A) \cdot P(B) \]

6. Bayes' Theorem

Bayes' Theorem provides a powerful means for updating the probability of a hypothesis as more evidence becomes available. The theorem is expressed as:
\[ P(S_i | E) = \frac{P(E | S_i) \cdot P(S_i)}{P(E)} \]
Where S_i represents different states of nature and E is an evidence outcome. This theorem assists in decision-making, especially in ambiguous probabilistic environments (Koller & Friedman, 2009).

7. Practical Applications of Probability

The applications of probability extend beyond academia into various fields, including finance, healthcare, and artificial intelligence. For example, businesses employ probability models to predict market trends and consumer behavior, while healthcare professionals use conditional probabilities to assess patient risks (Gensini et al., 2019).

8. Conclusion

An understanding of probability and its models is essential for rational decision-making in the face of uncertainty. By leveraging probability theory, practitioners can model real-world phenomena, assess outcomes, and make informed decisions backed by statistical evidence. As we continue to harness data in various domains, the relevance of probability will undoubtedly remain critical.

References

1. Blitzstein, J. K., & Hwang, J. (2015). Introduction to Probability. Chapman and Hall/CRC.
2. DeGroot, M. H., & Schervish, M. J. (2012). Probability and Statistics. Addison-Wesley.
3. Gensini, G. F., et al. (2019). Areas of interest in probabilistic models and statistics. European Journal of Internal Medicine, 62, e41-e42.
4. Hogg, R. V., & Tanis, E. A. (2001). Probability and Statistical Inference. Prentice Hall.
5. Koller, D., & Friedman, N. (2009). Probabilistic Graphical Models: Principles and Techniques. MIT Press.
6. McClave, J. T., & Sincich, T. (2018). Statistics. Pearson.
7. Mendenhall, W., Beaver, R. J., & Beaver, B. M. (2019). Introduction to Probability and Statistics. Cengage Learning.
8. Schafer, J. L. (2015). Analysis of Incomplete Multivariate Data. CRC Press.
9. Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: heuristics and biases. Science, 185(4157), 1124-1131.
10. Vardeman, S. B., & Jobe, J. (2014). Statistical Modeling: A Fresh Approach. Wiley.
By associating definitions with practical illustrations, this exposition serves as a foundational view into the complexities and utilities of probability within analytical contexts.