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Ch. 4 - Probability
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 4 - ProbabilityProblem 4.2.32
Chapter 4, Problem 4.2.32

Same Birthdays If 25 people are randomly selected, find the probability that no 2 of them have the same birthday. Ignore leap years.

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Step 1: Understand the problem. We are tasked with finding the probability that no two people in a group of 25 have the same birthday. This is a classic 'birthday problem' in probability, and we assume there are 365 days in a year (ignoring leap years).
Step 2: Recognize that this is a problem of permutations. If no two people share the same birthday, the first person can have any of the 365 days as their birthday, the second person can have any of the remaining 364 days, the third person can have any of the remaining 363 days, and so on, until the 25th person.
Step 3: Write the total number of ways to assign birthdays to 25 people without restriction. Since each person can have any of the 365 days, the total number of possible birthday assignments is \( 365^{25} \).
Step 4: Write the number of favorable outcomes where no two people share the same birthday. This is given by the product \( 365 \times 364 \times 363 \times \ldots \times (365 - 24) \), which can also be written as \( \frac{365!}{(365 - 25)!} \).
Step 5: Calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes. The probability is \( P = \frac{365 \times 364 \times 363 \times \ldots \times (365 - 24)}{365^{25}} \). Simplify this expression to find the final probability.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it helps quantify the chance that no two people among a group share the same birthday. Understanding basic probability principles, such as the total number of outcomes and favorable outcomes, is essential for solving the problem.
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Combinatorics

Combinatorics is a branch of mathematics dealing with counting, arrangement, and combination of objects. In the birthday problem, it is used to calculate the number of ways to assign unique birthdays to each person in a group. This involves understanding permutations and combinations, which are crucial for determining the total possible arrangements of birthdays.

Complementary Events

Complementary events are pairs of outcomes where one event occurs if and only if the other does not. In this scenario, instead of directly calculating the probability that no two people share a birthday, it can be easier to calculate the probability that at least two people do share a birthday and subtract that from 1. This approach simplifies the calculations and provides a clearer path to the solution.
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Related Practice
Textbook Question

Phone Numbers Current rules for telephone area codes allow the use of digits 2–9 for the first digit, and 0–9 for the second and third digits, but the last two digits cannot both be 1 (to avoid confusion with area codes such as 911). How many different area codes are possible with these rules? That same rule applies to the exchange numbers, which are the three digits immediately preceding the last four digits of a phone number. Given both of those rules, how many 10-digit phone numbers are possible? Given that these rules apply to the United States and Canada and a few islands, are there enough possible phone numbers? (Assume that the combined population is about 400,000,000.)

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Textbook Question

In Exercises 9–20, use the data in the following table, which lists survey results from high school drivers at least 16 years of age (based on data from “Texting While Driving and Other Risky Motor Vehicle Behaviors Among U.S. High School Students,” by O’Malley, Shults, and Eaton, Pediatrics, Vol. 131, No. 6). Assume that subjects are randomly selected from those included in the table. Hint: Be very careful to read the question correctly.

Drinking and Driving If one of the high school drivers is randomly selected, find the probability of getting one who drove when drinking alcohol.

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Textbook Question

Simulating Dice When two dice are rolled, the total is between 2 and 12 inclusive. A student simulates the rolling of two dice by randomly generating numbers between 2 and 12. Does this simulation behave in a way that is similar to actual dice? Why or why not?

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Textbook Question

Computer Variable Names A common computer programming rule was that names of variables must be between one and eight characters long. The first character can be any of the 26 letters, while successive characters can be any of the 26 letters or any of the 10 digits. For example, allowable variable names include A, BBB, and M3477K. How many different variable names are possible? (Ignore the difference between uppercase and lowercase letters.)

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Textbook Question

In Exercises 33–40, use the given probability value to determine whether the sample results are significant.



Voting Repeat Exercise 33 after replacing 40 Democrats being placed on the first line of voting ballots with 27 Democrats being placed on the first line. The probability of getting a result as high as 27 is 0.029792.

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Textbook Question

In Exercises 33–40, use the given probability value to determine whether the sample results are significant.



Selfie Deaths Based on Priceonomics data describing 49 deaths while taking selfies, it was found that 37 of those deaths were males. Assuming that males and females are equally likely to have selfie deaths, there is a 0.000235 probability of getting 37 or more males. Is the result of 37 males significantly low, significantly high, or neither? Does the result suggest that male selfie deaths are more likely than female selfie deaths?

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