- Use the formula \(P(B \mid C) = \frac{n(B \cap C)}{n(C)}\).
- Extract values from the table: \(n(B \cap C) = 9\) and \(n(C) = 24\).
- Calculate the ratio: \(P(B \mid C) = \frac{9}{24} \approx 0.38\).
- The final answer is \(\boxed{P(B \mid C) = \frac{9}{24} = 0.38}\).
Explanation
- Understanding conditional probability in a two-way table
To find the conditional probability \(P(B \mid C)\), we use the definition of conditional probability from a contingency table. The conditional probability of event B given event C is defined as the number of outcomes that satisfy both B and C, divided by the total number of outcomes in category C.
The formula is:
\[P(B \mid C) = \frac{n(B \cap C)}{n(C)}\]
where \(n(B \cap C)\) is the count for the intersection of row B and column C, and \(n(C)\) is the total count for column C.
- Identifying values from the table
Looking at the provided table, we can identify the necessary values:
- The intersection of row B and column C is the value in the cell (B, C), which is \(9\). So, \(n(B \cap C) = 9\).
- The total for column C is given as \(24\). So, \(n(C) = 24\).
Substituting these values into our formula:
\[P(B \mid C) = \frac{9}{24}\]
- Simplifying and converting to decimal
Now, we simplify the fraction \(\frac{9}{24}\). Both the numerator and the denominator are divisible by \(3\):
\[\frac{9 \div 3}{24 \div 3} = \frac{3}{8}\]
To express this as a decimal, we perform the division:
\[3 \div 8 = 0.375\]
Rounding this to two decimal places, as shown in the options, gives us \(0.38\). Thus, \(P(B \mid C) = \frac{9}{24} = 0.38\).
Examples
Conditional probability is used in everyday decision-making, such as predicting the likelihood of an outcome given certain conditions. For instance, a hospital might use a two-way table to analyze patient outcomes (recovered vs. not recovered) based on the treatment received (Drug A vs. Drug B). By calculating the conditional probability of recovery given a specific drug, doctors can determine which treatment is more effective. If Drug A resulted in 15 recoveries out of 24 patients, the probability of recovery given Drug A is \(\frac{15}{24} = 0.625\), helping them make informed medical choices.