The Senate Consists Of 100 Senators, Of Whom 34 Are Republic
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.
In the given scenario, the likelihood of the bill passing depends on the proportion of senators supporting it within each party and the overall composition of the Senate. To analyze this, we will construct a probability tree to compute the probability that the bill passes, considering the marginals and conditionals of political affiliation and support for the bill.
Paper For Above instruction
Introduction
The passage of legislation in the Senate is often influenced by the partisan composition and the support levels among different political groups. This paper applies probabilistic methods, specifically probability trees, to understand the likelihood that a bill will pass given the composition and support trends among Senators. By computing the joint, marginal, and conditional probabilities, it becomes possible to model and predict legislative outcomes more accurately.
Constructing the Probability Tree
The initial step involves defining the different stages at which decisions or events occur: party affiliation and support for the bill. The probabilities for each branch are derived from the given data.
- Probability that a randomly selected senator is Republican: \( P(R) = \frac{34}{100} = 0.34 \)
- Probability that a senator is Democrat: \( P(D) = \frac{66}{100} = 0.66 \)
Within each party, the likelihood of supporting the bill:
- Support among Republicans: \( P(Support|R) = 0.70 \)
- Support among Democrats: \( P(Support|D) = 0.35 \)
Correspondingly, the probabilities of not supporting:
- \( P(Not Support|R) = 1 - 0.70 = 0.30 \)
- \( P(Not Support|D) = 1 - 0.35 = 0.65 \)
The total probability that a senator supports the bill (pivotal for passing):
\[
P(Support) = P(R) \times P(Support|R) + P(D) \times P(Support|D) = (0.34)(0.70) + (0.66)(0.35) = 0.238 + 0.231 = 0.469
\]
- The probability of opposition:
\[
P(Not Support) = 1 - 0.469 = 0.531
\]
To determine if the bill passes, it must secure a simple majority: more than 50 senators. Since the expected number of supporters is approximately \( 0.469 \times 100 = 46.9 \), the probability the actual number exceeds 50 can be modeled using the binomial distribution or approximated through normal distribution. However, for our case, the probabilistic approach considers the simpler estimation based on support percentages.
Given the support probability, the key is whether more than 50 senators will support, which depends on the random selection and support probabilities.
Calculations Using the Probability Tree
The probability that the bill passes (that is, at least 51 or more senators support) requires calculating cumulative probabilities, which are complex under simple support probabilities. For simplicity, assume the number of supporters follows a binomial distribution with parameters \(n=100\) and \(p=0.469\).
The approximate probability that 51 or more support the bill involves the cumulative probability:
\[
P(X \geq 51) = 1 - P(X \leq 50)
\]
with \(X \sim Binomial(100, 0.469)\).
Using normal approximation for the binomial distribution:
- Mean: \(\mu = np = 100 \times 0.469 = 46.9 \)
- Variance: \(\sigma^2 = np(1-p) = 100 \times 0.469 \times 0.531 \approx 24.9\)
- Standard deviation: \(\sigma \approx \sqrt{24.9} \approx 4.99\)
Applying continuity correction:
\[
P(X \geq 51) \approx P\left(Z \geq \frac{50.5 - 46.9}{4.99}\right) = P(Z \geq 0.74)
\]
where \( Z \) is standard normal.
From standard normal tables:
\[
P(Z \geq 0.74) \approx 0.2296
\]
Thus, the approximate probability that the bill will pass is about 22.96%.
Conclusion
While the probabilistic model suggests that there is roughly a 23% chance for the bill to pass based on the current support levels and Senate composition, this is a simplified estimation. A more precise calculation could involve direct binomial probability summation, but the normal approximation provides a practical estimate. This analysis showcases how probability trees and distributions can be used to predict legislative outcomes with uncertain support.
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