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

The probability that no one enters the casino between 12:00 and 12:05 is approximately 0.0821 or 8.21%. The probability that at least 4 people enter the casino during that time is approximately 0.2425 or 24.25%.

Step 1: 1. Calculate the average rate of events per time interval (λ) for the given time frame (12:00 to 12:05).

As people enter the casino at a rate of 1 every 2 minutes, we need to calculate the total number of people expected to enter the casino in 5 minutes (from 12:00 to 12:05). Since 1 person enters every 2 minutes, in 5 minutes, we would expect: \[\lambda = \frac{1 \, \text{person}}{2 \, \text{minutes}} \times 5 \, \text{minutes} = 2.5 \, \text{people}\]

Step 2: 2. Part (a): Calculate the probability that no one enters between 12:00 and 12:05.

To find the probability that no one enters the casino between 12:00 and 12:05, we need to find the Poisson probability for \(k = 0\), using the average rate of events per time interval (\(\lambda = 2.5\)) calculated in Step 1. \[P(0; 2.5) = \frac{e^{-2.5} \cdot 2.5^0}{0!} = \frac{e^{-2.5} \cdot 1}{1} = e^{-2.5} \approx 0.0821\] So, the probability that no one enters the casino between 12:00 and 12:05 is approximately 0.0821 or 8.21%.

Step 3: 3. Part (b): Calculate the probability that at least 4 people enter the casino during that time.

To find the probability that at least 4 people enter the casino from 12:00 to 12:05, we first calculate the probabilities for exactly 4, 5, 6, etc., people entering the casino, and then find the sum of these probabilities. However, an efficient way to approach this is to find the complementary probability (i.e., probability that fewer than 4 people enter the casino), and then subtract this value from 1. We already have the probability for 0 people entering the casino from part (a). Now, we need to calculate the probabilities for 1, 2, and 3 people entering the casino, using the same formula as before: \[P(1; 2.5) = \frac{e^{-2.5} \cdot 2.5^1}{1!} \approx 0.2052\] \[P(2; 2.5) = \frac{e^{-2.5} \cdot 2.5^2}{2!} \approx 0.2565\] \[P(3; 2.5) = \frac{e^{-2.5} \cdot 2.5^3}{3!} \approx 0.2137\] Now, we find the sum of these probabilities: \(P(0) + P(1) + P(2) + P(3) \approx 0.0821 + 0.2052 + 0.2565 + 0.2137 = 0.7575\) The complementary probability is the probability that fewer than 4 people enter the casino. To find the probability that at least 4 people enter the casino, we subtract this value from 1: \[1 - (P(0) + P(1) + P(2) + P(3)) = 1 - 0.7575 = 0.2425\] So, the probability that at least 4 people enter the casino during the given time is approximately 0.2425 or 24.25%.

Key Concepts

Probability Theory

Probability theory is the branch of mathematics that deals with quantifying the likelihood of events. It provides a numerical representation of how likely it is for an event to occur, ranging from 0 (impossible event) to 1 (certain event). In the context of our exercise, the event is the entry of people into a casino over a specified period.

Mathematically, the probability of an event is often expressed as the ratio of the number of favorable outcomes to the total number of possible outcomes, assuming each outcome has an equal chance of occurring. For discrete distributions like the Poisson distribution used in the exercise, probabilities of specific outcomes are calculated using specific formulas. The Poisson distribution is particularly suited for modeling the number of times an event happens in a fixed interval of time or space when the events occur independently.

Rate of Events

The rate of events, commonly denoted by the symbol \( \lambda \) (lambda), is a key concept in Poisson probability distribution. It represents the average number of times an event occurs in a predefined interval. In our casino example, \( \lambda \) is defined as the average number of people entering the casino every 5 minutes.

Understanding and calculating the rate of events is crucial when modeling real-world processes using a Poisson distribution. It allows us to not only predict the probability of no events occurring within a set time frame, for instance, but also to determine the likelihood of multiple events. The proper assessment of the rate \( \lambda \) is essential for accurate probability calculations in various practical scenarios, from customer arrivals to system failures.

Complementary Probability

Complementary probability is a concept in probability theory that involves the likelihood of the opposite of a particular event. The complementary probability of an event A is denoted as \( P(A^c) \) and is calculated as \( 1 - P(A) \) where \( P(A) \) is the probability of the event occurring. This becomes a powerful tool when the direct calculation of the desired probability is difficult.

For example, in the casino situation, finding the probability of at least 4 people entering the casino directly by summing the probabilities of 4, 5, 6,... arrivals is cumbersome. Instead, by calculating the probability of the complementary event (fewer than 4 arrivals), we simplify the task. We then subtract this probability from 1 to get the probability of at least 4 arrivals—highlighting why understanding complementary probability is vital for efficient problem solving.

Factorial Representation

Factorial representation is a mathematical concept often denoted by an exclamation mark (!). For any non-negative integer \( n \), the factorial \( n! \) is defined as the product of all positive integers less than or equal to \( n \). In practical terms, it expresses the number of ways in which \( n \) objects can be arranged.

In the calculation of Poisson probabilities, factorials appear in the denominator as part of the distribution's formula. This is because the Poisson formula is derived from probability functions that involve counting distinct arrangements where the order of events doesn't matter. Understanding factorial representation is essential, as miscalculating \( n! \) will lead to incorrect probabilities. For students working with textbook problems, recognizing the significance of factorials and their correct computation in probability functions will enhance their problem-solving accuracy.