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Solve Poisson Probability Problem: Find P(0) with μ=5 Using Poisson Formula
Statistics
University Grade 1
Question Content
Let $P(x)=\\frac{\\mu^x \\cdot e^{-\\mu}}{x!}$ and let $\\mu=5$. Find $P(0)$. Round to six decimal places as needed.
Correct Answer
0.006738
Detailed Solution Steps
1
Step 1: Identify the given values and the target value. We are given the Poisson probability formula $P(x)=\\frac{\\mu^x \\cdot e^{-\\mu}}{x!}$, $\\mu=5$, and we need to calculate $P(0)$ which means $x=0$.
2
Step 2: Substitute the values into the formula. Replace $\\mu$ with 5 and $x$ with 0: $P(0)=\\frac{5^0 \\cdot e^{-5}}{0!}$.
3
Step 3: Apply the rules of exponents and factorials. Recall that any non-zero number to the power of 0 is 1, so $5^0=1$. Also, the factorial of 0, $0!$, is defined as 1. The formula simplifies to $P(0)=\\frac{1 \\cdot e^{-5}}{1}=e^{-5}$.
4
Step 4: Calculate the numerical value. $e^{-5}$ is approximately 0.006737947. Round this to six decimal places, which gives 0.006738.
Knowledge Points Involved
1
Poisson Probability Formula
The Poisson probability formula $P(x)=\\frac{\\mu^x \\cdot e^{-\\mu}}{x!}$ is used to calculate the probability of a given number of events ($x$) occurring in a fixed interval of time or space, where $\\mu$ is the average rate of occurrence, $e$ is the base of the natural logarithm (approximately 2.71828), and $x!$ is the factorial of $x$. It is commonly applied in scenarios where events happen independently at a constant average rate, such as the number of customer arrivals at a store per hour or the number of machine malfunctions per day.
2
Zero Exponent Rule
The zero exponent rule states that for any non-zero real number $a$, $a^0=1$. This rule simplifies calculations involving exponents when the exponent is 0, as seen when substituting $x=0$ into the Poisson formula where $5^0=1$.
3
Factorial of Zero
The factorial of 0, denoted as $0!$, is defined to be 1. This definition is established to maintain consistency in combinatorial and probability formulas, especially in cases where we need to calculate arrangements of zero items, which is considered a single possible arrangement.
4
Rounding Decimal Numbers
Rounding decimal numbers involves adjusting a number to a specified number of decimal places. When rounding, look at the digit immediately after the desired decimal place: if it is 5 or greater, round up the last digit; if it is less than 5, keep the last digit the same. In this problem, we round $e^{-5} \\approx 0.006737947$ to six decimal places, resulting in 0.006738.
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