Math IA - The Birthday Paradox
“What is the probability that at least 2 people in a room of 30 random people will have the same birthday?”
Probability is always surrounding us from stock markets to the ever-simple heads or tails. This very complicated area of mathematics can be explained in a simpler way. It is how likely an event is to happen. The probability of an event will always be between 0 and 1. The closer it is to one, the more likely the event is to happen.
I chose this topic because when I first read the birthday problem in the textbook, I tried to solve it repeatedly but each time I would get a very low probability. After re reading the question for the 20th time I finally realized my error. I was considering the probability that people would have the same birthday as me when the question was focusing on the probability that anyone in the room has the same birthday. When I finally managed to solve the question I didn’t know what else this could be used for so I did more research. That is when I discovered that the birthday paradox can also be used to crack hashing algorithms and can be used in cryptography.
The square below represents the sample space (all possible outcomes)
We have not calculated the probability yet so lets assume that the white represents the number of people who have the same birthday. We will call this P(sb).
The gray area represents the number of people who have different birthdays. We will call this P(db).
The reason we use this formula in particular is because the birthdays of 30 random people in a room are all independent events that do not rely on one another for it’s occurrence.
Since the maximum probability of an event is 1 this means:
P(db) + P(sb) = 1
This in turn means:
P(sb) = 1 - P...
... middle of paper ...
...tal part of lives just like privacy. Using cryptology provides mechanisms through a digital signature. This signature is inserted using a key (that only the writer of the email possesses) whilst a timestamp binds itself to the document. This type of cryptography is used to control access of security installations or pay-per-view television channels.
This IA started out with me trying to solve a question that I had tried to solve for ages and now I have realized that we ignore some amazing mathematics because we are so self-involved. The real birthday problem is not the question itself but it is a test of our ignorance. My research and this problem solving have helped be more objective with my thinking and understand how amazing probability really is.
I definitely want to solve more of these types of questions and learn more about the other fields of mathematics.
My fifteenth birthday was celebrated like any other, as was my sixteenth. They were both filled with joy, as I realized that even with the weight of culture on my shoulders, it was who I spent my coming of age with, not where I was or what it signified. I was surrounded by friends and family that love and care for me; who know that age is just a number. It’s my character that defines my maturity, and they looked at me no lesser than if my birthdays had been filled with frills and ceremony. In fact, they may think better of
The Lottery of Birth is discussed as the ideas, beliefs, religion, and culture we are born into. The creators of the film believe as though people do not question the belief systems they are born into, instead they thoughtlessly adopt the belief systems they were raised to know. I agree with the point that you are born into certain belief systems and that many people do not question it, but I also believe that you have the choice to change what you believe if you disagree with it. If you were born into a certain religion but do not particularly agree with the ideas or belief of that religion I think that you have the choice and the right to convert or change to a different religion. The same goes with patriotism, many people have moved from
When someone tells you traits about a person, and then gives you a list of possible work fields/hobbies that particular person might be in, it’s easy to pick and choose what you think is likely and what is not. But what happens when they mix two, a likely with an unlikely? The fact is, it becomes even more unlikely. The chance of choosing two work fields/hobbies of a person is always going to be less likely than just one. Though many seem to jump to the conclusion that because the unlikely is tied to a likely trait then it must be more likely.
Siegel explains the usage of it and how different cultures have adapted it for their uses. The Chinese based their version of the year of people’s birth while Europeans based it on locations of the Sun, Moon and planets connected to different Zodiac signs (Siegel 2014). Nowadays, people turn to the Zodiac Table in finding some guidance into how to live their lives. Since the Zodiac table has been an idea accepted for a long time and is ingrained in many various cultures in the world, some people do not question its validity. As explained earlier just because Zodiac signs seem reasonable, it does not make Astrology a science to
I calculated three probabilities in Benchmark 2. I did this in one way I used the same way for all three, but all three scenarios were different. For the first scenario there was one action play, one sound device, one conic device that repels zombies and one work schedule. To find the probability of this outcome I had to times the conic device and the sound device’s fraction together because they're the same. The fractions are 13/100. After I multiplied this fraction I added all three of the fractions together, which were16910,000+829+425. Adding all of these together I got .45276, which gives us a 45% chance of survival for this example. For the second example I took two action plans, one sound device, one conic device that hides you from
The aim of the person was a great factor in differentiating between the theoretical and experimental probability. According to our data (experimental) from our photograph poster , which had about 240 datum in it, said that Paris was the most hit photograph with 14.3 % of the pennies landing there, and the Disneyworld landed at second with 9.1% of the pennies landing on it. Similarly, in our theoretical data, the probability of a penny landing on Paris was the greatest at 7.3%/. On the other hand, our theoretical data also showed that landing on Disney World was very unlikely, at 4.4%. The fact that a lot of the pennies ended up landing in the Disney World photograph shows that people were likely aiming for that particular photograph because
The audio that I listened called Stochasticity, says a lot about human inferences and slippery evidence. Slippery evidence can affect human inferences. What this is saying is that at first we may infer or think one thing, but as we look further we see that something else is true. The audio guides us through three main stories, a very lucky wind, seeking patterns, and random rules. The first story, a very lucky wind was about a girl named Laura, who wrote on a balloon “please return to Laura Buxton.” After letting the balloon go, the balloon lands just 140 miles south in the yard of a girl named Laura Buxton also. You could say at first that it was fate that the balloon landed there, but it was actually just chance. The second story was about
Birthday Party, a short story written by Katherine Bush in 1946 explores the topic of marriage and its repercussions. Katherine Bush showcases the idea of an individual's involvement in an unloving and abusive marriage lead to the victims lead to despair and unhappiness. The theme of the short story is developed through the use of extended metaphors. The wife, who is the victim of the husband’s verbal abuse attempts to hide her sadness from the outside world and her “big hat” is a metaphor for this statement. Similar to how no one else is unaware of her sadness, the big hat covers her face and she goes unnoticed by many. The narrator states that she witnesses the woman “[c]rying quietly and heartbrokenly and hopelessly, all to herself, under the gay brim of her
Let’s say that a year is always 365 days long. The chance that the second classmate has the same birthday is 1/365. To find the probability that both people have the same birthday, we must multiply their separate probabilities…
Throughout the book no one is celebrated individually even when it comes to birthdays no one knows there personal birthday. Instead they would have a ceremony off all the Nines, children who are turning nine years old, and celebrate everyone at once. Additionally, everyone in the community dressed the same, for the young girls they were required to wear their hair neatly up in a ponytail along with a bow.
...tional expense) goes up as well. (pg. 87) There is proof for this in two phenomena, “dependent events” and “statistical fluctuations.” (pg. 87) Dependent event is an outcome that is affected by other, usually previous outcomes. An example is removing colored crayons from a bag. Each time you remove a crayon the chances of drawing out a certain color will change. (pg. 88) Statistical fluctuations are fluctuations derived from many random processes. The more times the process is done, the higher the chance of more standard statistical fluctuation. An example of this is when a fair coin is tossed many times and the number of counted heads and tails is about the same leading to a ratio very close to 1. After only a few throws, outcomes with a drastic difference of heads over tails or vice versa is possible, however the more throws, the closer to a 1 to 1 ratio. (pg. 88)
As a society, we hold dear the value of privacy and confidentiality as a basic fundamental right. Access to the new technology of electronic mail can jeopardize both values.
Cryptography is such a broad part of our lives we do not even notice the smallest applications, shopping on eBay or watching satellite television. I bet you even used cryptology when you were in school and did not even know it. Ever write a message in numbers instead of letters? Each letter of the alphabet correlated to its number position in the alphabet. The number sequence 3,16,25,12,20,15,12,15,7,25 equals cryptology. This is a form of cryptology in its most basic form. Webster's dictionary defines Cryptography as :
When we wanted to read any Ba Zi chart, there had to be a starting point or reference point. That starting point was the day master. Everything else in the birth chart was read in relation to this day master.
These counting principles are important in understanding odds. For example, I rarely play the lottery but have relatives who play on a regular basis. Mathematically the odds are slim, however the game can only be won by chance. I hope the definitions and the equations have brought clarity to the subject.