The Venn diagram provided shows a sample space and two...

The Venn diagram provided shows a sample space and two events $A$ and $B$. Suppose $P(a)-0.32, P(b)-0.17, P(c)-0.28,$ and $P(d)-0.23 .$ Confirm that the probabilities of the outcomes add up to 1 , then compute the following probabilities. a. $P(A)$. b. $P(B)$. c. $P\left(A^{c}\right)$ two ways: (i) by finding the outcomes in $A^{C}$ and adding their probabilities, and (ii) using the Probability Rule for Complements. d. $P(A \cap B)$ e. $P(A \cup B)$ two ways: (i) by finding the outcomes in $A \cup B$ and adding their probabilities, and (ii) using the Additive Rule of Probability.

The Venn diagram provided shows a sample space and two events $A$ and $B$. Suppose $P(a)-0.32, P(b)-0.17, P(c)-0.28,$ and $P(d)-0.23 .$ Confirm that the probabilities of the outcomes add up to 1 , then compute the following probabilities.
a. $P(A)$.
b. $P(B)$.
c. $P\left(A^{c}\right)$ two ways: (i) by finding the outcomes in $A^{C}$ and adding their probabilities,
and (ii) using the Probability Rule for Complements.
d. $P(A \cap B)$
e. $P(A \cup B)$ two ways: (i) by finding the outcomes in $A \cup B$ and adding their
probabilities, and (ii) using the Additive Rule of Probability.

Key Concepts

Sample Space

The sample space is the complete set of all possible outcomes of an experiment. In probability theory, the sum of the probabilities of all outcomes in the sample space must equal 1. This concept is fundamental as it ensures that you have accounted for every possible scenario in the analysis.

Venn Diagrams

A Venn diagram is a visual tool used to represent events as sets, showing their relationships such as intersections, unions, and complements. It is particularly useful in probability for illustrating how events overlap and for organizing information to compute probabilities accurately.

Probability of an Event

The probability of an event is computed by summing the probabilities of all outcomes that constitute that event. This approach relies on the principle of additivity for mutually exclusive outcomes, ensuring that every part of the event is considered without double counting.

Complement Rule

The complement rule states that the probability of the complement of an event (not occurring) is equal to one minus the probability of the event itself. This is expressed as P(A^c) = 1 - P(A) and provides an alternative way to calculate probabilities when it is easier to analyze the complementary set.

Intersection of Events

The intersection of two events represents the outcomes that are common to both events. In probability, P(A ? B) is used to quantify the likelihood that both events occur simultaneously. This concept is crucial when analyzing cases where events are not mutually exclusive.

Union of Events and Additive Rule

The union of two events includes all outcomes that belong to either one of the events or both. The additive rule for probability expresses this as P(A ? B) = P(A) + P(B) - P(A ? B). This formula ensures that any overlap between the events is not counted twice, providing an accurate computation of the overall probability of either event occurring.

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