Suppose that we have a sample space S={E1, E2, E3, E4, E5, E6, E7}, where E1, E2, …, E7 denote t

step 1 Here first the probability of A will be probability of E1 plus probability of E4 plus probabilit

Suppose that we have a sample space S={E1, E2, E3, E4, E5,...

Suppose that we have a sample space $S=\left\{E_1, E_2, E_3, E_4, E_5, E_6, E_7\right\}$, where $E_1, E_2, \ldots, E_7$ denote the sample points. The following probability assignments apply: $P\left(E_1\right)=.05, P\left(E_2\right)=20, P\left(E_3\right)=.20, P\left(E_4\right)=.25, P\left(E_5\right)=.15, P\left(E_6\right)=.10$, and $P\left(E_7\right)=05$. Let $$ \begin{aligned} & A=\left\{E_1, E_4, E_6\right\} \\ & B=\left\{E_2, E_4, E_7\right\} \\ & C=\left\{E_2, E_3, E_5, E_1\right\} \end{aligned} $$ a. Find $P(A), P(B)$, and $P(C)$. b. Find $A \cup B$ and $P(A \cup B)$. c. Find $A \cap B$ and $P(A \cap B)$. d. Are events $A$ and $C$ mutually exclusive? c. Find $B^c$ and $P\left(B^{\circ}\right)$.

Suppose that we have a sample space $S=\left\{E_1, E_2, E_3, E_4, E_5, E_6, E_7\right\}$, where $E_1, E_2, \ldots, E_7$ denote the sample points. The following probability assignments apply: $P\left(E_1\right)=.05, P\left(E_2\right)=20, P\left(E_3\right)=.20, P\left(E_4\right)=.25, P\left(E_5\right)=.15, P\left(E_6\right)=.10$, and $P\left(E_7\right)=05$. Let
$$
\begin{aligned}
& A=\left\{E_1, E_4, E_6\right\} \\
& B=\left\{E_2, E_4, E_7\right\} \\
& C=\left\{E_2, E_3, E_5, E_1\right\}
\end{aligned}
$$
a. Find $P(A), P(B)$, and $P(C)$.
b. Find $A \cup B$ and $P(A \cup B)$.
c. Find $A \cap B$ and $P(A \cap B)$.
d. Are events $A$ and $C$ mutually exclusive?
c. Find $B^c$ and $P\left(B^{\circ}\right)$.

Key Concepts

Mutually Exclusive Events

Mutually exclusive events are those that cannot occur at the same time, implying that their intersection is empty. If two events are mutually exclusive, the probability of their union is simply the sum of their individual probabilities.

Intersection of Events

The intersection of two events includes only the outcomes that are common to both events. It represents the scenario in which both events occur simultaneously, and its probability is derived from the probabilities of the overlapping outcomes.

Union of Events

The union of two (or more) events consists of all outcomes that belong to at least one of the events. Calculating the probability of a union typically involves adding the probabilities of individual events and subtracting the probabilities of their intersections to avoid double counting when the events are not mutually exclusive.

Complement of an Event

The complement of an event consists of all outcomes in the sample space that are not part of the event. It represents the occurrence of 'not the event' and is used to calculate probabilities where the event does not happen by subtracting the event’s probability from one.

Probability Assignments

Probability assignments involve associating a probability value with each outcome in the sample space. These values must be nonnegative and sum to 1 for the entire sample space. They represent how likely each outcome is to occur.

Sample Space

The sample space is the set of all possible outcomes in an experiment. In probability theory, every event is a subset of this space. It forms the foundation upon which probabilities are assigned to various outcomes or collections of outcomes.

Event

An event is defined as any subset of the sample space, consisting of one or more outcomes. Events can represent simple occurrences (a single outcome) or more complex ones, and they are the primary objects for which probabilities are calculated.

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