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Chapter 1: Problem 10

For a bill to come before the president of the United States, it must be passed by both the House of Representatives and the Senate. Assume that, of the bills presented to these two bodies, 60 percent pass the House, 80 percent pass the Senate, and 90 percent pass at least one of the two. Calculate the probability that the next bill presented to the two groups will come before the president.

Short Answer

Expert verified
The probability is 0.50.

Step by step solution

01

Define the Variables

Let \( H \) be the event that a bill passes the House of Representatives, and \( S \) be the event that a bill passes the Senate. We know \( P(H) = 0.60 \), \( P(S) = 0.80 \), and \( P(H \cup S) = 0.90 \). We need to calculate \( P(H \cap S) \), the probability that a bill passes both the House and the Senate.
02

Apply the Union Formula

We use the formula for the probability of the union of two events: \[ P(H \cup S) = P(H) + P(S) - P(H \cap S) \]Plug in the known values: \[ 0.90 = 0.60 + 0.80 - P(H \cap S) \]
03

Solve for \(P(H \cap S)\)

Simplify the equation: \[ 0.90 = 1.40 - P(H \cap S) \]Rearrange to find \(P(H \cap S)\): \[ P(H \cap S) = 1.40 - 0.90 = 0.50 \]
04

Interpret the Probability

The probability that a bill will pass both the House and the Senate, and therefore come before the president, is \(0.50\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Union of Events
In probability, understanding the union of events is a basic yet vital concept. The union of two events, denoted as \( H \cup S \), signifies that either event \( H \), event \( S \), or both occur.
This forms the foundation for the probability of a union of events. The formula used in this context is:
  • \( P(H \cup S) = P(H) + P(S) - P(H \cap S) \)
The union formula helps us handle any overlap between events. It subtracts the overlap, \( P(H \cap S) \), to avoid double-counting, providing the correct total probability.
In our example, with the House and Senate passing rates known, the union helps determine the overlapping probability. This union insight drives the calculation of the bill reaching the president's desk. Understanding this not only helps in mathematics but also gets us one step closer to deciphering real-world probabilities.
Conditional Probability
Conditional probability examines the likelihood of an event occurring, given that another event has already happened. It is a vital concept when understanding interconnected occurrences. Denoted as \( P(A | B) \), it represents the probability of event \( A \) given event \( B \) has occurred.In the context of political processes, consider: if a bill passes the House, what is the probability it also passes the Senate? Although not directly calculated in our example, this would be solved as:
  • \( P(S | H) = \frac{P(H \cap S)}{P(H)} \)
This analysis helps us understand the dependencies between certain events and improves our predictive accuracy in interconnected systems, such as political decision-making.
Grasping conditional probability opens doors to deeper insights into how events influence each other, both theoretically and practically.
Probability in Politics
Believe it or not, probability plays a significant role in politics. The process a bill undergoes before reaching the president is a clear example. By studying probabilities in these contexts, we can form more precise predictions about political decisions.For instance, with known probabilities that a bill passes individual houses, politicians and analysts predict the bill's pathway.
  • \( P(H) = 0.60 \) for House.
  • \( P(S) = 0.80 \) for Senate.
  • \( P(H \cap S) = 0.50 \), combined passage.
Understanding these outcomes helps strategize political moves, estimate passing rates, and calculate risks effectively. Probability guides political strategies, election forecasts, and policy impacts. A well-grounded comprehension of probability ensures more informed and strategic political decisions, affecting policymaking and governance on a broader scale.
Step-by-Step Solutions
Breaking a problem into step-by-step solutions aids in decomposition, making complex issues easier to tackle. This methodically addresses aspects of a problem, ensuring nothing is overlooked and provides clarity.The provided solution follows these steps:
  • Defining variables: Assigning \( H \) for House and \( S \) for Senate events.
  • Applying formulas: Utilizing the union of events principle.
  • Solving equations: Isolating and calculating \( P(H \cap S) \).
  • Interpreting results: Understanding the probability of a comprehensive outcome.
This process transforms daunting problems into manageable tasks. Adopting a step-by-step approach can build confidence, ensure accuracy, and reinforce understanding in students.

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Most popular questions from this chapter

Here is an attempt to get around the fact that we cannot choose a "random integer." (a) What, intuitively, is the probability that a "randomly chosen" positive integer is a multiple of \(3 ?\) (b) Let \(P_{3}(N)\) be the probability that an integer, chosen at random between 1 and \(N,\) is a multiple of 3 (since the sample space is finite, this is a legitimate probability). Show that the limit $$ P_{3}=\lim _{N \rightarrow \infty} P_{3}(N) $$ exists and equals \(1 / 3\). This formalizes the intuition in (a), and gives us a way to assign "probabilities" to certain events that are infinite subsets of the positive integers. (c) If \(A\) is any set of positive integers, let \(A(N)\) mean the number of elements of \(A\) which are less than or equal to \(N\). Then define the "probability" of \(A\) as $$ P(A)=\lim _{N \rightarrow \infty} A(N) / N $$ provided this limit exists. Show that this definition would assign probability 0 to any finite set and probability 1 to the set of all positive integers. Thus, the probability of the set of all integers is not the sum of the probabilities of the individual integers in this set. This means that the definition of probability given here is not a completely satisfactory definition. (d) Let \(A\) be the set of all positive integers with an odd number of digits. Show that \(P(A)\) does not exist. This shows that under the above definition of probability, not all sets have probabilities.

Our calendar has a 400-year cycle. B. H. Brown noticed that the number of times the thirteenth of the month falls on each of the days of the week in the 4800 months of a cycle is as follows: Sunday 687 Monday 685 Tuesday 685 Wednesday 687 Thursday 684 Friday 688 Saturday 684 From this he deduced that the thirteenth was more likely to fall on Friday than on any other day. Explain what he meant by this.

A die is rolled until the first time that a six turns up. We shall see that the probability that this occurs on the \(n\) th roll is \((5 / 6)^{n-1} \cdot(1 / 6)\). Using this fact, describe the appropriate infinite sample space and distribution function for the experiment of rolling a die until a six turns up for the first time. Verify that for your distribution function \(\sum_{\omega} m(\omega)=1\).

Let \(X\) be a random variable with distribution function \(m_{X}(x)\) defined by $$ m_{X}(-1)=1 / 5, \quad m_{X}(0)=1 / 5, \quad m_{X}(1)=2 / 5, \quad m_{X}(2)=1 / 5 $$ (a) Let \(Y\) be the random variable defined by the equation \(Y=X+3\). Find the distribution function \(m_{Y}(y)\) of \(Y\). (b) Let \(Z\) be the random variable defined by the equation \(Z=X^{2}\). Find the distribution function \(m_{Z}(z)\) of \(Z\).

For a bill to come before the president of the United States, it must be passed by both the House of Representatives and the Senate. Assume that, of the bills presented to these two bodies, 60 percent pass the House, 80 percent pass the Senate, and 90 percent pass at least one of the two. Calculate the probability that the next bill presented to the two groups will come before the president.
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