No Excel spreadsheets, please. I need to see the work to better understand the material. **Understanding the Birthday Paradox**The text explores the probability problem known as the birthday paradox. It challenges our intuitive understanding of probabilities by investigating how likely it is for at least two people in a room to share the same birthday.### Example ProblemConsider a room with 25 people.1. **Observation on Probability**: - *Question*: Do you think the probability that at least two people share a birthday (same month and day) is above or below 50%? Would you say the probability is below 10% or above 90%? Explain your reasoning. *Answer*: The intuitive guess for most people might be that it's less likely for two people to share a birthday. However, due to the birthday paradox, the probability is counterintuitively high.2. **Smallest Group Size**: - *Question*: If we assume there are 365 days in a year, what is the smallest number of people in a room that would guarantee at least two people share a birthday? *Answer*: To guarantee sharing a birthday, we need 366 people (one more than the number of days in a year). This assures at least one shared birthday by the pigeonhole principle.### Detailed ExplanationWith just two people in a room, the probability can be calculated directly. Defining the event of two people not sharing a birthday:\[ P(\text{Different Birthday}) = \frac{365}{365} \times \frac{364}{365} \]This represents the first person having a birthday and the second person not having the same birthday as the first.Now, the complementary event (at least one shared birthday) is:\[ P(\text{Same Birthday}) = 1 - P(\text{Different Birthday}) = 1 - \frac{365 \times 364}{365 \times 365} = 1 - \frac{364}{365} \approx 0.0027 \]So with two people, the probability that they share a birthday is approximately 0.27%.With three people, the probability calculation adapts to:\[ P(\text{At Least Two People with the Same Birthday}) = 1 - \frac{365 \times 364 \times 363}{365^3} \]### Further Examination3. **Completing the Probability Example**: - *Question*: Complete the computation above. Did the probability increase or decrease by adding just one extra person

1) The objective is to given reasoning whether the probability of at least two people sharing a…

It is often the case that we can gauge how likely an event is by simply thinking of our past experiences and comparing it to other events in our daily lives. For instance, most people would agree that the probability of buying one lottery ticket and winning the jackpot is much lower than the probability of rolling one die and getting a six, which is 1 in 6. We could use data or computations to confirm our intuition about these situations. In fact, if a lottery was set up where you choose 6 numbers (in any order) from a possible pool of 49 numbers, your chances of winning the jackpot are 1 in 13,983,816. Sometimes, however, our intuition betrays us. In this activity, we will investigate a classic probability problem called the birthday paradox. Consider a room that has 25 people in it. 1. Do you think that the probability of at least two people sharing a birthday (same month and same day) is above or below 50%? Would you say the probability is below 10% or above 90%? Explain your reasoning. 2. If we assume that there are 365 days in a year, what is the smallest number of people in a room that would guarantee that at least two people in the room share a birthday? If we have two people in a room, they either share a birthday or they don't. These are complementary events, and we can write the following equation. P(Same Birthday)=1-P(Different Birthday) Since there are 365 days in a year, there are 365(364)=132,860 different ways that the two people can have different birthdays. There are also 365(365)=133,225 possible pairs of birthdays. From this information, we get the following probability. Different Birthdays 365(364) P(Same Birthday)=1- =1- 0.0027 All Possible Birthdays 365 (365) Hence, with two people in a room, the probability that they share a birthday is 0.27%. In a room with three people, we can use the same argument. Either no birthday is shared or at least two people share a birthday. We would have the following probability: P(At Least Two People with the Same Birthday)= 365(364) (363) 365(365) (365) 3. Complete the computation above. Did the probability increase or decrease by adding just one extra person? By how much?

It is often the case that we can gauge how likely an event is by simply thinking of our past experiences and comparing it to other events in our daily lives. For instance, most people would agree that the probability of buying one lottery ticket and winning the jackpot is much lower than the probability of rolling one die and getting a six, which is 1 in 6. We could use data or computations to confirm our intuition about these situations. In fact, if a lottery was set up where you choose 6 numbers (in any order) from a possible pool of 49 numbers, your chances of winning the jackpot are 1 in 13,983,816. Sometimes, however, our intuition betrays us. In this activity, we will investigate a classic probability problem called the birthday paradox. Consider a room that has 25 people in it. 1. Do you think that the probability of at least two people sharing a birthday (same month and same day) is above or below 50%? Would you say the probability is below 10% or above 90%? Explain your reasoning. 2. If we assume that there are 365 days in a year, what is the smallest number of people in a room that would guarantee that at least two people in the room share a birthday? If we have two people in a room, they either share a birthday or they don't. These are complementary events, and we can write the following equation. P(Same Birthday)=1-P(Different Birthday) Since there are 365 days in a year, there are 365(364)=132,860 different ways that the two people can have different birthdays. There are also 365(365)=133,225 possible pairs of birthdays. From this information, we get the following probability. Different Birthdays 365(364) P(Same Birthday)=1- =1- 0.0027 All Possible Birthdays 365 (365) Hence, with two people in a room, the probability that they share a birthday is 0.27%. In a room with three people, we can use the same argument. Either no birthday is shared or at least two people share a birthday. We would have the following probability: P(At Least Two People with the Same Birthday)= 365(364) (363) 365(365) (365) 3. Complete the computation above. Did the probability increase or decrease by adding just one extra person? By how much?

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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