Problem 63 - A First Course in Probability

Question

People enter a gambling casino at a rate of 1 every 2 minutes. (a) What is the probability that no one enters between 12: 00 and \(12: 05 ?\) (b) What is the probability that at least 4 people enter the casino during that time?

Step-by-Step Solution

Answer

(a) The probability that no one enters between 12:00 and 12:05 is approximately \(0.0821\). (b) The probability that at least 4 people enter the casino during that time is approximately \(0.2446\).

Step 1: Calculate the rate for the given time interval

As mentioned earlier, we first calculate the rate for the given interval by multiplying the given rate by the length of the interval in minutes. In this case, λ = 5 * (1/2) = 2.5 people.

Step 2: Use the Poisson formula to calculate the probability

Recall that the Poisson formula is: \[P(X=k) = (\exp(-\lambda) * \lambda^k) / k!\] Where X is the number of events (people entering the casino), k is the desired number of events (in this case, 0), λ is the rate (2.5 people per 5-minute interval), and k! is the factorial of k. Now we plug in the values to find the probability of no one entering between 12:00 and 12:05: \[P(X=0) = (\exp(-2.5) * 2.5^0) / 0! = \exp(-2.5) = 0.0821\] For (b):

Step 1: Calculate probabilities for 0, 1, 2, and 3 people entering

We use the same Poisson formula to find the probability of 0, 1, 2, and 3 people entering the casino between 12:00 and 12:05. \[P(X=0) = 0.0821 \text{ (from part a)}\] \[P(X=1) = (\exp(-2.5) * 2.5^1) / 1! = 0.2043\] \[P(X=2) = (\exp(-2.5) * 2.5^2) / 2! = 0.2554\] \[P(X=3) = (\exp(-2.5) * 2.5^3) / 3! = 0.2136\]

Step 2: Subtract the probabilities of 0, 1, 2, and 3 people entering from 1 to find the probability of at least 4 people entering

Now we add the probabilities of 0, 1, 2, and 3 people entering together and subtract that value from 1 to find the probability of at least 4 people entering the casino during the 5-minute interval: \[P(X \geq 4) = 1 - (P(X=0) + P(X=1) + P(X=2) + P(X=3))\] \[P(X \geq 4) = 1 - (0.0821 + 0.2043 + 0.2554 + 0.2136) = 1 - 0.7554 = 0.2446\] So the probability of no one entering between 12:00 and 12:05 is approximately 0.0821, and the probability of at least 4 people entering the casino during that time is approximately 0.2446.

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that is concerned with analyzing random phenomena and events. It provides a framework for predicting the likelihood of different outcomes. In probability theory, the likelihood of an event occurring is expressed as a value between 0 and 1, with 0 indicating impossibility and 1 indicating certainty.

In the context of the Poisson distribution, we focus on counting the number of events happening within a fixed interval of time or space. For example, if we want to find out the probability of people entering a casino within a specific time frame, we use Poisson distribution to calculate it.

Understanding probability involves recognizing that each outcome has a certain likelihood of occurring, which can be expressed through mathematical formulas. Such insights help us make informed decisions based on the probabilities of different scenarios.

Exponential Function

The exponential function is a key mathematical function that appears frequently in probability theory and statistics. It is denoted by the symbol \( e^x \), where \( e \) is a constant approximately equal to 2.71828. This function describes growth or decay processes that occur continuously and compounding at every moment in time.

In the Poisson distribution, the exponential function is crucial for determining the probability of rare events. For example, when we want to calculate the probability that no people enter a casino in a given time frame, we use the exponential function as part of the Poisson formula: \( P(X=k) = (\exp(-\lambda) \cdot \lambda^k) / k! \). Here, \( \lambda \) represents the average rate of occurrence within an interval, and \( \exp(-\lambda) \) is derived from the exponential function.

This function simplifies complex processes and helps us determine the probability of various outcomes accurately. By using the exponential function, we can model situations in a precise mathematical manner.

Factorial Calculation

Factorial calculation is an important concept when dealing with discrete probability distributions like the Poisson distribution. The factorial of a number is the product of all positive integers less than or equal to that number, and it is denoted by \( k! \). For example, the factorial of 3, written as \( 3! \), is \( 3 \times 2 \times 1 = 6 \).

Factorials are used in probability formulas because they help distribute probabilities over multiple possible outcomes. In the Poisson distribution formula, \( P(X=k) = (\exp(-\lambda) \cdot \lambda^k) / k! \), the factorial term, \( k! \), in the denominator plays a crucial role in adjusting the probability. It ensures that the calculation reflects the different ways an event can happen.

Understanding how to calculate factorials accurately is essential in probability theory as it impacts the final probability value that we are aiming to find. Factorials can seem tricky at first, but with practice, they become an easy yet powerful tool to comprehend random processes.