Confirm that the probabilities in the two-way contingency...

Confirm that the probabilities in the two-way contingency table add up to $1,$ then use it to find the probabilities of the events indicated. $$ \begin{array}{|c|c|c|c|} \hline & R & S & T \\ \hline M & 0.09 & 0.25 & 0.19 \\ \hline N & 0.31 & 0.16 & 0.00 \\ \hline \end{array} $$ a. $P(R), P(S), P(R \cap S)$ b. $P(M), P(N), P(M \cap N)$ C. $P(R \cup S)$. d. $P\left(R^{c}\right)$ e. Determine whether or not the events $N$ and $S$ are mutually exclusive; the events $N$ and $T$.

Confirm that the probabilities in the two-way contingency table add up to $1,$ then use it to find the probabilities of the events indicated.
$$
\begin{array}{|c|c|c|c|}
\hline & R & S & T \\
\hline M & 0.09 & 0.25 & 0.19 \\
\hline N & 0.31 & 0.16 & 0.00 \\
\hline
\end{array}
$$
a. $P(R), P(S), P(R \cap S)$
b. $P(M), P(N), P(M \cap N)$
C. $P(R \cup S)$.
d. $P\left(R^{c}\right)$
e. Determine whether or not the events $N$ and $S$ are mutually exclusive; the events $N$ and $T$.

Key Concepts

Mutually Exclusive Events

Mutually exclusive events are events that cannot occur at the same time; their intersection is empty, meaning the probability of both occurring together is zero. In a contingency table, if two events are mutually exclusive, the cell corresponding to their intersection will have a probability of zero, confirming that the events have no overlap.

Complementary Probability

Complementary probability pertains to the likelihood that an event does not occur. It is calculated as one minus the probability of the event. This concept is useful for finding probabilities indirectly by subtracting the probability of the event from one, particularly when its direct computation might be more complex.

Union of Events

The union of two events represents the occurrence of at least one of the events. Its probability is calculated using the principle of inclusion?exclusion, which states that the probability of the union equals the sum of the probabilities of each event minus the probability of their intersection. This accounts for any overlap that might have been counted twice.

Intersection of Events

The intersection of two events refers to the occurrence of both events simultaneously. In a contingency table, the probability of the intersection is given directly by the value in the cell at the intersection of the respective row and column, indicating that both conditions are met.

Marginal Probability

Marginal probability is the probability of an event occurring irrespective of the outcome of another variable. In the context of a contingency table, marginal probabilities are obtained by summing the probabilities across a row (or column). They provide the likelihood of one variable’s outcome without considering the other variable.

Two-Way Contingency Table

A two?way contingency table is a tabular representation of the joint probabilities (or frequencies) of two categorical variables. Each cell in the table represents the probability of a specific combination of the two variables’ outcomes. The table allows one to easily compute various probabilities by summing across rows (for one variable) or columns (for the other variable) and check that the overall total probability sums to one.

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