Questions: If a reporter stops a random senator after the vote, what is the probability that the senator will either be a Republican or will have voted against the bill? Express your answer as a simplified fraction or a decimal. Do not round any intermediate calculations.

$If a reporter stops a random senator after the vote, what is the probability that the senator will either be a Republican or will have voted against the bill? Express your answer as a simplified fraction or a decimal. Do not round any intermediate calculations.$

Transcript text: \begin{tabular}{|c|c|c|} \hline & Voted in Favor & Voted Against \\ \hline Republican & 29 & 14 \\ \hline Democrat & 39 & 12 \\ \hline Other & 1 & 5 \\ \hline \end{tabular} If a reporter stops a random senator after the vote, what is the probability that the senator will either be a Republican or will have voted against the bill? Express your answer as a simplified fraction or a decimal. Do not round any intermediate calculations.

Solution

Solution Steps

To find the probability that a randomly selected senator is either a Republican or has voted against the bill, we can use the principle of inclusion-exclusion. First, calculate the total number of senators. Then, find the number of Republicans and the number of senators who voted against the bill. Finally, subtract the overlap (Republicans who voted against) to avoid double-counting, and divide by the total number of senators to get the probability.

Step 1: Calculate Total Number of Senators

The total number of senators is the sum of all those who voted in favor and against, across all parties: \[ \text{Total Senators} = 29 + 14 + 39 + 12 + 1 + 5 = 100 \]

Step 2: Calculate Number of Republicans

The number of Republicans is the sum of those who voted in favor and against: \[ \text{Republicans} = 29 + 14 = 43 \]

Step 3: Calculate Number of Senators Who Voted Against

The number of senators who voted against the bill is: \[ \text{Voted Against} = 14 + 12 + 5 = 31 \]

Step 4: Calculate Number of Republicans Who Voted Against

The number of Republicans who voted against the bill is: \[ \text{Republicans Against} = 14 \]

Step 5: Apply Inclusion-Exclusion Principle

To find the probability that a senator is either a Republican or voted against, use the inclusion-exclusion principle: \[ P(\text{Republican or Voted Against}) = \frac{\text{Republicans} + \text{Voted Against} - \text{Republicans Against}}{\text{Total Senators}} \] Substitute the values: \[ P(\text{Republican or Voted Against}) = \frac{43 + 31 - 14}{100} = \frac{60}{100} = 0.6 \]