Extra Credit 3

Due: November 9, 2023
Points: 30

The birthday problem asks how many people must be in a room so that the probability of two of them having the same birthday is 0.5. This problem has you explore it by simulation. Basically, you will create a series of lists of random numbers of length n = 2, …, and look for duplicates. You will do this 5000 times for each length. For each length, count the number of lists with at least 1 duplicate number; then divide that number by 5000. That is the (simulated) probability that a list of n generated numbers has at least one duplicate. As the random numbers you generate are between 1 and 365 (each one corresponding to a day of the year), this simulates the birthday problem.

Now, breathe deeply and calm down. We will do this in steps!

First, detecting duplicates. Write a function called hasduplicates(l) that takes a list l and returns True if it contains a duplicate element, and False if it does not. For example:
```
>>> hasduplicates([1, 2, 3, 4, 5, 5, 2])↵
True
>>> hasduplicates([1, 2, 3, 4, 5, 6, 7])↵
False
```
Now, deal with one set of birthdays. Write a function called onetest(count) that generates a list of count random integers between 1 and 365 inclusive, and returns True if it contains a duplicate element, and False if it does not. Please use the function hasduplicates(l) to test for duplicates.
Now for the probability for count people. Write a function probab(count, num) that runs num tests of count people, and counts the number of tests with duplicates. It returns the fraction of the tests with duplicates; that is, the number of duplicates divided by num.
Now for the demonstration. Start with 2 people, and begin adding people until the probability of that many people having two people with a birthday in common is over 0.9. (In other words, start with a list of 2 elements, and increase the number of elements in the list until the simulation shows a probability of 0.9 that a number in the list is duplicated.) Print each probability; your output should look like this:
```
For  2 people, the probability of 2 birthdays is 0.00220
For  3 people, the probability of 2 birthdays is 0.00880
For  4 people, the probability of 2 birthdays is 0.01680
For  5 people, the probability of 2 birthdays is 0.02940
For  6 people, the probability of 2 birthdays is 0.03940
For  7 people, the probability of 2 birthdays is 0.05900
For  8 people, the probability of 2 birthdays is 0.06840
For  9 people, the probability of 2 birthdays is 0.09700
For 10 people, the probability of 2 birthdays is 0.12360
```
How many people are needed so that the probability of two of them with a birthday in common is over 0.9? How many are needed such that the probability of two of them having the same birthday is at least 0.5? Put these answers into a comment at the head of the file.
Hint: Don’t be surprised if your probabilities are slightly different than the ones shown in the sample output. As randomness is involved, it is very unlikely your numbers will match the ones shown here.

To turn in: Please call your program bday.py and submit it to Canvas

Matt Bishop
Office: 2209 Watershed Sciences
Phone: +1 (530) 752-8060
Email: [email protected]

ECS 235A, Computer and Information Security
Version of October 30, 2023 at 10:13PM

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