Consider The Following Joint Probability Table

Consider The Following Joint Probability Tableb1b2b3b4a0130110180

Consider the following joint probability table. B1 B2 B3 B4 A 0....10 Ac 0....17 a. What is the probability that A occurs? (Round your answer to 2 decimal places.) Probability: b. What is the probability that B2 occurs? (Round your answer to 2 decimal places.) Probability: c. What is the probability that Ac and B4 occur? (Round your answer to 2 decimal places.) Probability: d. What is the probability that A or B3 occurs? (Round your answer to 2 decimal places.) Probability: e. Given that B2 has occurred, what is the probability that A occurs? (Round your intermediate calculations and final answers to 4 decimal places.) Probability: f. Given that A has occurred, what is the probability that B4 occurs? (Round your intermediate calculations and final answers to 4 decimal places.) Probability:

Paper For Above instruction

In the realm of probability theory, understanding joint, marginal, and conditional probabilities is fundamental to analyzing complex probabilistic systems. Given the joint probability table involving variables B1, B2, B3, B4, and A, we can deduce various probabilities that help describe the likelihood of events occurring individually or in conjunction. This paper explores these probabilistic calculations step by step, demonstrating their applications and importance in statistical analysis.

Introduction to Probabilistic Concepts

Joint probability refers to the likelihood of two or more events occurring simultaneously. For instance, in the given table, each cell represents the probability of a specific combination of B1, B2, B3, B4, and A. Marginal probability involves summing over certain variables to find the probability of a single event, regardless of others. Conditional probability measures the likelihood of an event given that another event has already occurred, calculated as the ratio of the joint probability of both events to the probability of the known event. Mastery of these concepts allows analysts to make inferences, predictions, and denote the dependence or independence between variables.

Analysis of the Joint Probability Table

Suppose the table provides probabilities for various combinations of B1, B2, B3, B4, and A, with specific probabilities assigned to each. The sum of all probabilities in the table equals 1, adhering to the fundamental property of probability distributions. The given data points such as 0.10 and 0.17 are likely the probabilities associated with specific combinations or marginal sums, though the complete table must be considered for precise calculations.

Calculating the Probabilities

a. Probability that A occurs

To find P(A), sum all joint probabilities where A occurs, i.e., sum over all B1, B2, B3, B4 where A is present. Mathematically:

P(A) = Σ P(A, B1, B2, B3, B4)

Using the data, sum all relevant probabilities from the table; for example, if the total sum of probabilities where A occurs equals 0.45, then the probability is 0.45 to two decimal places.

b. Probability that B2 occurs

Similarly, P(B2) is obtained by summing all probabilities where B2 occurs, regardless of other variables:

P(B2) = Σ P(B1, B2, B3, B4, A)

If the sum of such probabilities is 0.40, then the probability is 0.40.

c. Probability that Ac and B4 occur

This refers to the joint probability of the complement of A (not A) and B4:

P(A^c ∩ B4) = sum of all probabilities where A is not present and B4 occurs

Calculate by summing the respective probabilities from the table; for example, if the sum is 0.12, then the probability is 0.12.

d. Probability that A or B3 occurs

To find P(A ∪ B3), use the inclusion-exclusion principle:

P(A ∪ B3) = P(A) + P(B3) - P(A ∩ B3)

Calculate each component by summing relevant probabilities, then combine accordingly. For instance, if P(A) = 0.45, P(B3) = 0.50, and P(A ∩ B3) = 0.20, then:

P(A ∪ B3) = 0.45 + 0.50 - 0.20 = 0.75

e. Given that B2 has occurred, the probability that A occurs

This is a conditional probability:

P(A | B2) = P(A ∩ B2) / P(B2)

Calculate numerator by summing probabilities where both A and B2 occur; determine the denominator as P(B2). For example, if P(A ∩ B2) = 0.15 and P(B2) = 0.40, then:

P(A | B2) = 0.15 / 0.40 = 0.3750

f. Given that A has occurred, the probability that B4 occurs

Similarly,

P(B4 | A) = P(A ∩ B4) / P(A)

Compute these probabilities from the table; if P(A ∩ B4) = 0.20 and P(A) = 0.45, then:

P(B4 | A) = 0.20 / 0.45 ≈ 0.4444

Conclusion

Calculating various probabilities from a joint probability table allows us to understand the relationships between different variables. The process involves summing relevant entries for marginal and joint probabilities and applying principles such as inclusion-exclusion and conditional probability formulas. These calculations are fundamental in fields such as statistics, machine learning, and risk analysis, where understanding dependencies among variables greatly impacts inference and decision-making.

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