Understanding and Calculating Conditional Probability of Complementary Events

This problem involves calculating the conditional probability of the complement of event A given the complement of event B, using the provided probabilities of events A, B, and the conditional probability of A given B. We are given the following probabilities:

• \({\rm{P}}\left( {\rm{A}} \right) = \frac{3}{5}\)
• \({\rm{P}}\left( {\rm{B}} \right) = \frac{3}{{10}}\)
• \({\rm{P}}\left( {{\rm{A|B}}} \right) = \frac{2}{3}\)

We need to find the value of \({\rm{P}}\left( {{{\rm{\overline A }}}{\rm{|}}{{\rm{\overline B }}}} \right)\), where A̅ and B̅ are the complementary events of A and B respectively.

Formula for Conditional Probability

The conditional probability of event X occurring given that event Y has occurred is defined as:

\(P(X|Y) = \frac{P(X \cap Y)}{P(Y)}\), provided \(P(Y) > 0\).

Using this formula, we can write the expression for \({\rm{P}}\left( {{{\rm{\overline A }}}{\rm{|}}{{\rm{\overline B }}}} \right)\) as:

\(P(\overline{A} | \overline{B}) = \frac{P(\overline{A} \cap \overline{B})}{P(\overline{B})}\)

Finding the Components

Step 1: Calculate \(P(A \cap B)\)

We are given \(P(A|B)\) and \(P(B)\). We know that \(P(A|B) = \frac{P(A \cap B)}{P(B)}\). We can rearrange this formula to find \(P(A \cap B)\):

\(P(A \cap B) = P(A|B) \times P(B)\)

Substitute the given values:

\(P(A \cap B) = \frac{2}{3} \times \frac{3}{10}\)

\(P(A \cap B) = \frac{2 \times 3}{3 \times 10} = \frac{6}{30} = \frac{1}{5}\)

Step 2: Calculate \(P(A \cup B)\)

The probability of the union of two events A and B is given by the formula:

\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

Substitute the given values for \(P(A)\), \(P(B)\), and the calculated value for \(P(A \cap B)\):

\(P(A \cup B) = \frac{3}{5} + \frac{3}{10} - \frac{1}{5}\)

To add/subtract these fractions, find a common denominator, which is 10:

\(P(A \cup B) = \frac{3 \times 2}{5 \times 2} + \frac{3}{10} - \frac{1 \times 2}{5 \times 2}\)

\(P(A \cup B) = \frac{6}{10} + \frac{3}{10} - \frac{2}{10}\)

\(P(A \cup B) = \frac{6 + 3 - 2}{10} = \frac{7}{10}\)

Step 3: Calculate \(P(\overline{A} \cap \overline{B})\)

Using De Morgan's Law, we know that \(\overline{A} \cap \overline{B} = \overline{A \cup B}\). The probability of the complement of an event E is \(P(\overline{E}) = 1 - P(E)\).

Therefore, \(P(\overline{A} \cap \overline{B}) = P(\overline{A \cup B}) = 1 - P(A \cup B)\).

Substitute the calculated value for \(P(A \cup B)\):

\(P(\overline{A} \cap \overline{B}) = 1 - \frac{7}{10}\)

\(P(\overline{A} \cap \overline{B}) = \frac{10}{10} - \frac{7}{10} = \frac{10 - 7}{10} = \frac{3}{10}\)

Step 4: Calculate \(P(\overline{B})\)

The probability of the complement of event B is \(P(\overline{B}) = 1 - P(B)\).

Substitute the given value for \(P(B)\):

\(P(\overline{B}) = 1 - \frac{3}{10}\)

\(P(\overline{B}) = \frac{10}{10} - \frac{3}{10} = \frac{10 - 3}{10} = \frac{7}{10}\)

Step 5: Calculate \(P(\overline{A} | \overline{B})\)

Now we have both parts needed for the conditional probability formula \(P(\overline{A} | \overline{B}) = \frac{P(\overline{A} \cap \overline{B})}{P(\overline{B})}\).

Substitute the calculated values for \(P(\overline{A} \cap \overline{B})\) and \(P(\overline{B})\):

\(P(\overline{A} | \overline{B}) = \frac{\frac{3}{10}}{\frac{7}{10}}\)

Dividing by a fraction is the same as multiplying by its reciprocal:

\(P(\overline{A} | \overline{B}) = \frac{3}{10} \times \frac{10}{7}\)

\(P(\overline{A} | \overline{B}) = \frac{3 \times 10}{10 \times 7} = \frac{30}{70} = \frac{3}{7}\)

Thus, the probability of the complement of A given the complement of B is \(\frac{3}{7}\).

Summary of Probabilities Calculated

Revision Table: Key Probability Concepts

Additional Information: Understanding Complementary and Conditional Probability

A complementary event \(\overline{E}\) is the event that E does not occur. If you consider the sample space (all possible outcomes), E and \(\overline{E}\) cover the entire sample space and are mutually exclusive. Their probabilities sum to 1, i.e., \(P(E) + P(\overline{E}) = 1\).

Conditional probability \(P(X|Y)\) changes the sample space to be just the outcomes where Y occurs. It represents the probability of X happening within this reduced sample space (where Y is guaranteed to have happened). The formula \(P(X|Y) = \frac{P(X \cap Y)}{P(Y)}\) formally captures this, essentially measuring the portion of Y that also includes X.

In this problem, we used the relationship between the intersection of complements and the complement of the union (\(\overline{A} \cap \overline{B} = \overline{A \cup B}\)). This is a fundamental concept derived from set theory and applicable to probability.

Calculating \(P(\overline{A} | \overline{B})\) involves finding the probability of the outcomes where neither A nor B happens, relative to the outcomes where B does not happen. Using De Morgan's law helps simplify the numerator \(P(\overline{A} \cap \overline{B})\) by relating it to \(P(A \cup B)\), which is easier to calculate using the inclusion-exclusion principle \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\).

This problem demonstrates how different probability rules and definitions can be combined to solve more complex problems involving conditional probabilities and complements.