/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 44 The number \(x\) of people enter... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 44

The number \(x\) of people entering the intensive care unit at a particular hospital on any one day has a Poisson probability distribution with mean equal to five persons per day. a. What is the probability that the number of people entering the intensive care unit on a particular day is two? Less than or equal to two? b. Is it likely that \(x\) will exceed \(10 ?\) Explain.

Short Answer

Expert verified
Answer: The probability of exactly 2 people entering the ICU on a particular day is approximately 8.42%. The cumulative probability of 2 or fewer people entering the ICU on a particular day is approximately 26.50%. The likelihood of more than 10 people entering the ICU is approximately 1.37%.

Step by step solution

01

Find P(X=2)#

Using the Poisson formula, with \(\lambda = 5\) and k=2, we get: P(X = 2) = \(\frac{e^{-5}(5)^2}{2!}\) Calculate this value: P(X = 2) 鈮� 0.0842
02

Find P(X鈮�2)#

To calculate the cumulative probability for X鈮�2, we will sum up the probabilities for X=0, 1, and 2: P(X鈮�2) = P(X=0) + P(X=1) + P(X=2) Using the Poisson formula for each term: P(X鈮�2) = \(\frac{e^{-5}(5)^0}{0!}\) + \(\frac{e^{-5}(5)^1}{1!}\) + \(\frac{e^{-5}(5)^2}{2!}\) Calculate this value: P(X鈮�2) 鈮� 0.2650
03

Reason about P(X>10)#

To analyze the probability of more than 10 people entering the ICU, we will calculate the complement event P(X鈮�10) and subtract it from 1: P(X>10) = 1 - P(X鈮�10) To calculate the cumulative probability for X鈮�10, we will sum up the probabilities for X=0, 1, ..., 10 using the Poisson formula: P(X鈮�10) = P(X=0) + P(X=1) + ... + P(X=10) Calculate this value using a calculator or software: P(X鈮�10) 鈮� 0.9863 Now, subtract it from 1 to get the probability of more than 10 people entering the ICU: P(X>10) = 1 - 0.9863 鈮� 0.0137
04

Interpret the Results#

a. The probability that the number of people entering the ICU on a particular day is 2 is approximately 0.0842 or 8.42%. The cumulative probability that 2 or fewer people enter the ICU on a particular day is approximately 0.2650 or 26.50%. b. The probability that more than 10 people enter the ICU on a particular day is approximately 0.0137 or 1.37%. Since this probability is low, it is unlikely that more than 10 people will enter the ICU on a given day.

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