The Senate Consists Of 100 Senators Of Whom 34 Are Republica
12 The Senate Consists Of 100 Senators Of Whom 34 Are Republicans A
The Senate consists of 100 senators, of whom 34 are Republicans and 66 are Democrats. A bill to increase defense appropriations is before the Senate. Thirty-five percent of the Democrats and 70% of the Republicans favor the bill. The bill needs a simple majority to pass. Using a probability tree, determine the probability that the bill will pass.
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The legislative process within the Senate often involves probabilistic assessments to understand the likelihood of bill passage, especially when party affiliations and voting tendencies influence the outcome. In this analysis, we will utilize a probability tree to determine the chance that a bill to increase defense appropriations will pass, considering the voting preferences of Senators based on party affiliation.
Understanding the Composition and Voting Preferences
The Senate comprises 100 Senators: 34 Republicans and 66 Democrats. Their preferences concerning the bill differ; 70% of Republicans favor it, while only 35% of Democrats do. Since the bill requires a simple majority—in this case, at least 51 Senators voting in favor—it's important to analyze the probabilities of various voting combinations leading to the bill’s passage.
Constructing the Probability Tree
The initial branch separates Senators based on party affiliation: Republican or Democrat. The probability that a randomly selected Senator is Republican is \( P(R) = \frac{34}{100} = 0.34 \), and Democrat is \( P(D) = \frac{66}{100} = 0.66 \).
Next, for each party, the probability that a randomly selected Senator favors the bill is:
- Republicans: \( P(\text{favor} | R) = 0.70 \)
- Democrats: \( P(\text{favor} | D) = 0.35 \)
Conversely, for each party, the probability that a senator does not favor the bill:
- Republicans: \( P(\text{not favor} | R) = 0.30 \)
- Democrats: \( P(\text{not favor} | D) = 0.65 \)
The probability tree captures all possible combinations of Senators favoring or opposing the bill. To find the probability that the bill passes, we need to assess the aggregate probability of getting at least 51 senators favoring the bill under the distribution derived from the binomial probabilities.
Calculating the Probabilities of Favoring Senators
Given the large number of Senators in each group, the binomial distribution provides an approximate way to calculate the probability that a certain number of Senators support the bill within each party.
For Republicans:
- Number of Republicans: 34
- Favoring Republicans: follows \( \text{Binomial}(n=34, p=0.70) \)
For Democrats:
- Number of Democrats: 66
- Favoring Democrats: follows \( \text{Binomial}(n=66, p=0.35) \)
Our goal is to compute:
\[
P(\text{bill passes}) = P(\text{total supporters} \geq 51)
\]
which involves summing joint probabilities over all combinations where the total number of supporters from both parties adds up to at least 51.
Distributions and Approximation
Due to the complexity of summing over all combinations, a normal approximation to the binomial distribution can be used for the large group sizes:
- Republicans: mean \( \mu_R = 34 \times 0.70 = 23.8 \), variance \( \sigma^2_R = 34 \times 0.70 \times 0.30 = 6.06 \)
- Democrats: mean \( \mu_D = 66 \times 0.35 = 23.1 \), variance \( \sigma^2_D = 66 \times 0.35 \times 0.65 = 14.91 \)
The total number of supporters \( T \) is approximately normal with
\[
\mu_T = \mu_R + \mu_D = 23.8 + 23.1 = 46.9
\]
\[
\sigma_T = \sqrt{6.06 + 14.91} = \sqrt{20.97} \approx 4.58
\]
Using the normal approximation:
\[
P(T \geq 51) \approx 1 - \Phi\left(\frac{50.5 - \mu_T}{\sigma_T}\right)
\]
where 50.5 is the continuity correction.
Calculating:
\[
z = \frac{50.5 - 46.9}{4.58} \approx \frac{3.6}{4.58} \approx 0.785
\]
Using standard normal tables:
\[
P(T \geq 51) \approx 1 - \Phi(0.785) \approx 1 - 0.783 = 0.217
\]
Conclusion
The probability that the bill will pass, given the current distribution of party membership and voting preferences, is approximately 21.7%. While these calculations are approximate, they provide a meaningful estimate for legislative analysts and policymakers considering the probability of bill approval based on existing voting patterns.
Implications for the Senate's Decision-Making
This quantitative assessment underscores the importance of bipartisan support and strategizing to sway a sufficient number of Senators in favor of legislation. It highlights how voting preferences combined with party composition critically influence legislative outcomes, emphasizing the role of political dynamics in policy enactment.
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