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Chapter 5: Problem 47

Of the first 44 presidents of the United States (George Washington through Barack Obama), two had the same birthday (Polk and Harding). Is this highly coincidental? Answer by finding the probability of at least one birthday match among 44 people.

Short Answer

Expert verified
The probability of at least one birthday match among the 44 presidents is approximately 0.927, indicating it's not highly coincidental.

Step by step solution

01

Understanding the Problem

We need to find the probability that at least one pair of presidents out of 44 share the same birthday. This involves calculating the complement probability that all 44 presidents have different birthdays first.
02

Calculate Total Birthday Combinations

Assuming there are 365 possible birthdays (ignoring leap years for simplicity), the total possible ways to assign different birthdays to each of the 44 presidents is \( 365^{44} \).
03

Calculate Different Birthdays Combination

To have all 44 presidents with different birthdays, the first president has 365 choices, the second one 364 (as one date is taken), the third 363, and so on down to 322 for the 44th president. Hence, there are \( 365 \times 364 \times \, ... \, \times 322 \) ways for this.
04

Compute Probability of All Different Birthdays

The probability that all 44 presidents have different birthdays is given by \( \frac{365}{365} \times \frac{364}{365} \times ... \times \frac{322}{365} \).
05

Calculate At Least One Match Probability

The probability of at least one birthday match is the complement of all different birthdays, so it's \( 1 - \left( \frac{365!}{(365-44)! \times 365^{44}} \right) \).
06

Simplify the Expression

Simplifying this expression using approximation methods or technology (as exact calculations are complex) gives us a probability of approximately 0.927.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Birthday Paradox
The Birthday Paradox is a counter-intuitive concept in probability that showcases how surprisingly few people are needed in a group for a higher likelihood that at least two people share a birthday. Even with 23 people, there's already about a 50% chance of a shared birthday. When you consider 44 people, such as in the United States president scenario, this probability increases significantly. The paradox highlights the difference between intuitive thinking and actual mathematical probabilities. In this problem, the two presidents sharing a birthday among 44 presidents is a great demonstration of how our intuitions can underestimate these probabilities. Despite seeming improbable, the Birthday Paradox makes it clear that birthday matches in a group are not as coincidental as they seem.
Complement Rule
The Complement Rule is a core concept in probability used to find the chance of an event by calculating the likelihood that the event does not occur, and then subtracting from one. This rule is especially handy when the direct calculation of an event is complex. For our presidential scenario, instead of trying to directly compute the probability of at least one birthday match among the 44 presidents, it鈥檚 easier to calculate the probability that no one shares a birthday. This involves:
  • Finding the probability that all presidents have different birthdays.
  • Subtracting this probability from 1 to find the chance of at least one match.
By using the complement rule, we efficiently determine the probability of shared birthdays, showcasing how powerful this mathematical tool is.
Permutation and Combination
Understanding permutations and combinations is crucial when dealing with any probability involving arrangements or selections. In the context of the Birthday Paradox, permutations are key since we are interested in the arrangement of birthdays. A permutation takes into account the order, which is why calculating different birthdays involves permutations. We started by assigning a birthday to one president and continued sequentially, reducing the available birthdays by one each time, reflecting the ordered assignment of birthdays. Using permutations in this way reveals how we can systematically approach probability problems. Though permutations might sound complex, they are just a way to think about the different ways to arrange items, such as birthdays, while considering the importance of order.
Probability Calculation
Calculating probability involves more than plugging numbers into formulas鈥攊t requires understanding the problem's aspects and applying the right concepts. In the presidents' birthday problem, the calculation steps are crucial:
  • Firstly, calculate all possible birthday arrangements using powers (i.e., 365 raised to the power of number of people).
  • Then, calculate the arrangements where everyone has different birthdays using the permutation approach.
  • Determine the probability that all individuals have different birthdays by dividing these permutations by total arrangements.
  • Finally, apply the complement rule to find the probability of at least one match.
Overall, this approach simplifies what can initially seem overwhelming, offering a structured method to tackle probability with confidence.

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Most popular questions from this chapter

Your teacher gives a truefalse pop quiz with 10 questions. a. Show that the number of possible outcomes for the sample space of possible sequences of 10 answers is 1024 . b. What is the complement of the event of getting at least one of the questions wrong? c. With random guessing, show that the probability of getting at least one question wrong is \(0.999 .\)

A 2007 study by the National Center on Addiction and Substance Abuse at Columbia University reported that for college students, the estimated probability of being a binge drinker was 0.50 for males and 0.34 for females. Using notation, express each of these as a conditional probability.

Every year the insurance industry spends considerable resources assessing risk probabilities. To accumulate a risk of about one in a million of death, you can drive 100 miles, take a cross country plane flight, work as a police officer for 10 hours, work in a coal mine for 12 hours, smoke two cigarettes, be a nonsmoker but live with a smoker for two weeks, or drink 70 pints of beer in a year (Wilson and Crouch, \(2001,\) pp. \(208-209\) ). Show that a risk of about one in a million of death is also approximately the probability of flipping 20 heads in a row with a balanced coin.

Since events of low probability eventually happen if you observe enough trials, a monkey randomly pecking on a typewriter could eventually write a Shakespeare play just by chance. Let's see how hard it would be to type the title of Macbeth properly. Assume 50 keys for letters and numbers and punctuation. Find the probability that the first seven letters that a monkey types are macbeth. (Even if the monkey can type 60 strokes a minute and never sleeps, if we consider each sequence of seven keystrokes as a trial, we would wait on the average over 100,000 years before seeing this happen!)

Larry Bird, who played pro basketball for the Boston Celtics, was known for being a good shooter. In games during \(1980-1982,\) when he missed his first free throw, 48 out of 53 times he made the second one, and when he made his first free throw, 251 out of 285 times he made the second one. a. Form a contingency table that cross tabulates the outcome of the first free throw (made or missed) in the rows and the outcome of the second free throw (made or missed) in the columns. b. For a given pair of free throws, estimate the probability that Bird (i) made the first free throw and (ii) made the second free throw. c. Estimate the probability that Bird made the second free throw, given that he made the first one. Does it seem as if his success on the second shot depends strongly, or hardly at all, on whether he made the first?
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