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Chapter 5: Problem 8

The following data were collected by counting the number of operating rooms in use at Tampa General Hospital over a 20 -day period: On three of the days only one operating room was used, on five of the days two were used, on eight of the days three were used, and on four days all four of the hospital's operating rooms were used. a. Use the relative frequency approach to construct a probability distribution for the number of operating rooms in use on any given day. b. Draw a graph of the probability distribution. c. Show that your probability distribution satisfies the required conditions for a valid discrete probability distribution.

Short Answer

Expert verified
\(P(X=1) = 0.15, P(X=2) = 0.25, P(X=3) = 0.40, P(X=4) = 0.20\). Graph the bars. Valid distribution.

Step by step solution

01

Calculate Relative Frequencies

First, add up the total number of days observed: \(3 + 5 + 8 + 4 = 20\) days. Each relative frequency is computed by dividing the frequency of each outcome by the total: \( P(X=1) = \frac{3}{20} = 0.15 \), \( P(X=2) = \frac{5}{20} = 0.25 \), \( P(X=3) = \frac{8}{20} = 0.40 \), \( P(X=4) = \frac{4}{20} = 0.20 \).
02

Construct Probability Distribution Table

Create an organized table showing the number of operating rooms (\(X\)) and their corresponding probabilities (\(P(X)\)) obtained from the previous step:| Number of Rooms (X) | Probability (P(X)) ||---------------------|--------------------|| 1 | 0.15 || 2 | 0.25 || 3 | 0.40 || 4 | 0.20 |
03

Graph the Probability Distribution

Using the table from Step 2, draw a bar graph where the x-axis represents the number of operating rooms (1 through 4) and the y-axis represents the probabilities. Each bar will have a height corresponding to its probability value: 0.15, 0.25, 0.40, and 0.20 respectively.
04

Verify Valid Probability Distribution Conditions

A valid probability distribution must satisfy two conditions: 1. All probabilities must be between 0 and 1. This is true for our probabilities: 0.15, 0.25, 0.40, and 0.20.2. The sum of all probabilities must equal 1. Check this by adding *together the probability values: \(0.15 + 0.25 + 0.40 + 0.20 = 1\). Since both conditions are satisfied, the distribution is valid.

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Key Concepts

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

Relative Frequency
Relative frequency is a fundamental concept in understanding probability distribution and represents the proportion of times an event occurs compared to the total number of observations. In our example, we determine the relative frequency for each scenario of operating rooms in use at Tampa General Hospital over a 20-day period.
For instance:
  • One operating room used on 3 days implies a relative frequency of \( \frac{3}{20} = 0.15 \).

  • Similarly, two operating rooms used on 5 days translates to a relative frequency of \( \frac{5}{20} = 0.25 \).

  • For eight days with three rooms, the relative frequency is \( \frac{8}{20} = 0.40 \).

  • Finally, four rooms used on 4 days results in \( \frac{4}{20} = 0.20 \).
These calculations form the basis for crafting a probability distribution. Each relative frequency directly turns into a probability for that specific case.
Discrete Probability
Discrete probability deals with scenarios where outcomes can be counted and are distinct. In this case, the number of operating rooms in use at the hospital is a discrete random variable because it can take only distinct integer values: 1, 2, 3, or 4.
For each outcome, we calculate a probability using relative frequency. Therefore, the probabilities for each potential number of operating rooms are:
  • \( P(X=1) = 0.15 \)

  • \( P(X=2) = 0.25 \)

  • \( P(X=3) = 0.40 \)

  • \( P(X=4) = 0.20 \)
This probability distribution tells us the likelihood of each distinct number of operating rooms being used. It is vital for discrete events, which helps in making informed decisions based on possible outcomes.
Probability Graph
A probability graph visually represents how probabilities are distributed across outcomes. For the operating rooms example, the probability graph is a bar chart that provides an easy-to-understand view.
On the x-axis, we place the number of operating rooms (1 through 4), while the y-axis shows the probability values corresponding to each outcome. The heights of the bars represent the probabilities:
  • 0.15 for 1 room being used

  • 0.25 for 2 rooms

  • 0.40 for 3 rooms

  • 0.20 for 4 rooms
Such graphs are incredibly useful for quickly assessing the probability of different events. They provide a clear, visual representation of where focus should lie; higher bars indicate more frequent events.
Probability Distribution Conditions
There are two key conditions to confirm a valid probability distribution, ensuring the rules of probabilities are met:
  • Non-negativity: Each probability must lie between 0 and 1. - In our example, each calculated probability \(0.15, 0.25, 0.40,\) and \(0.20\) satisfies this condition.

  • Total Probability is 1: The sum of all probabilities must equal to 1. - Adding them up: \(0.15 + 0.25 + 0.40 + 0.20 = 1\), confirming our distribution is valid.
By ensuring these conditions, we can be confident that the distribution appropriately represents the probabilities of the various scenarios occurring, thus supporting accurate statistical analysis.

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