Question

The Birthday Problem Two people enter a room and their birthdays (ignoring years) are recorded. a. Identify the nature of the simple events in \(S\). b. What is the probability that the two people have a specific pair of birthdates? c. Identify the simple events in event \(A\) : Both people have the same birthday. d. Find \(P(A)\). e. Find \(P\left(A^{c}\right)\).

Short answer

#f. Interpret the results.# #tag_title#Interpretation of Results #tag_content#The probabilities calculated above show the likelihood of different birthday scenarios between two people. The probability of both people having the same birthday is relatively low, at \(P(A) = \frac{365}{365 * 365}\). On the other hand, the probability of both people not having the same birthday is higher, with \(P(A^c) = \frac{365 * 365 - 365}{365 * 365}\). This implies that it is more likely for two people to have different birthdays than to share the same birthdate.

Step-by-step solution

Description of Simple Events

The simple events in \(S\) are all the possible combinations of birthdays between the two people. Since we are ignoring the years and only considering the dates, there are 365 possible birthdates for each person. Therefore, simple event space \(S\) contains \(365 \times 365\) pairs of birthdates. #b. What is the probability that the two people have a specific pair of birthdates?#

Probability of a Specific Pair of Birthdates

To calculate this probability, we need to find the probability of each person having a specific birthdate and multiply them. To find the proportion of the specific pair out of the total possible pairs, we will have: \(P(\text{specific pair}) = \frac{1}{365 * 365}\) #с. Identify the simple events in event \(A\) : Both people have the same birthday.#

Simple Events in Event \(A\)

The simple events in event \(A\) are the pairs of birthdays in which both people have the same birthdate. For each of the 365 possible birthdates, there's one pair where both people have that birthday. So, event \(A\) contains 365 simple events. #d. Find \(P(A)\).#

Probability of Both People Having the Same Birthday

The probability of both people having the same birthday can be calculated by dividing the number of simple events in event \(A\) by the total number of events in \(S\): \(P(A) = \frac{\text{Number of pairs with the same birthday}}{\text{Total number of possible pairs}}\) \(P(A) = \frac{365}{365 * 365}\) #e. Find \(P\left(A^{c}\right)\).#

Probability of Both People Not Having the Same Birthday

The event \(A^c\) represents the complementary event, where both people do not have the same birthday. To find the probability of this event, we can use the formula for complementary events: \(P(A^c) = 1 - P(A)\) \(P(A^c) = 1 - \frac{365}{365 * 365}\) \(P(A^c) = \frac{365 * 365 - 365}{365 * 365}\)

Key concepts

Birthday Problem

The Birthday Problem is a classic paradox in probability theory. It explores the likelihood that, in a group of people, some pair will share the same birthday. At first glance, people often underestimate the probability of shared birthdays due to the seemingly high number of potential birthdays (365 days in a year). However, in groups as small as 23 people, the probability of two people sharing a birthday is surprisingly high - about 50%. This phenomenon demonstrates the counterintuitive nature of probability theory and highlights how our intuitive sense of probability can sometimes be flawed.
In the exercise, the goal is to identify probabilities related to two people having either the same or different birthdays, emphasizing the simplicity of the problem when confined to just two individuals.

Simple Events

Simple events are the most basic outcomes that cannot be broken down further. In the context of the Birthday Problem, a simple event is defined as a specific pair of birthdates for the two individuals involved. Each person has 365 possible birthdays, considering birthdays with each consisting of a day of the year (ignoring leap years).

• This means that the sample space consists of 365 x 365 = 133,225 possible outcomes, with each combination representing a unique pair of birthdays for the two individuals.
• A simple event, for example, could be person A is born on January 1st, and person B is born on February 2nd.

Understanding these simple events is crucial when calculating probabilities, as they form the foundation upon which probability calculations are based.

Complementary Events

Complementary events represent outcomes that together encompass all possible outcomes of an event. In probability, if we have an event A, its complement A^c consists of everything not in A. For the Birthday Problem, if event A is both people having the same birthday, its complement A^c would be both people having different birthdays.
This relationship can be depicted mathematically as:
\[ P(A^c) = 1 - P(A) \]Using complementary events simplifies calculations because instead of directly computing the probability for non-matching birthdays, we calculate the probability of matching birthdays and subtract it from 1. This method of determining probabilities can often save time and reduce the potential for errors in complex probability problems.

Probability Calculation

Probability calculations involve determining the likelihood of a specific event within a defined sample space. In the Birthday Problem, the probability of both people sharing the same birthday is computed by comparing the number of favorable outcomes to the total number of outcomes.

• The number of favorable outcomes is 365 (one for each day where the birthdays match).
• The total number of outcomes, as previously identified, is 365 x 365 = 133,225.

Therefore, the probability is calculated as:\[ P(A) = \frac{365}{133,225} \]The calculations do not stop here. For the event where the individuals have different birthdays, we use the complementary probability:\[ P(A^c) = 1 - P(A) \]Ultimately, this exercise highlights the simplicity and elegance of probability calculation by leveraging fundamental concepts such as sample space and complementary events.