What is probability? Sure, we looked at where it started, but what is the math behind it?
Let's look at this video to get an introduction to probability.
So we introduced the Problem of Points but didn't really look at how to solve it. Let's see how Sal and Brit introduce the problem in the video below.
So why couldn't Sal just take the whole pot since he was so close to winning? Does Brit have a chance to win if they kept playing? Let's check out the next video
to see how they solved this problem.
So Cardano came up with this main idea of probability:
We also have these other formulas that are very useful in probability.
Let A and B be two events.
The probability range goes from 0 to 1, another gift from Cardano.
When adding probabilities, you add the probability of each event, but you need to make sure that you subtract the probability of both events occuring.
The rule of complementary events states that the probability of one event happening plus the probability of that event not happening, will be equal to 1.
If events are disjoint, then there is no probability that the events will occur at the same time.
An independent event means that one event can happen without relying on what happens in another event. If an event is dependent,
then what happens in a previous event, affects the other event. For example, finding the probability of drawing two Aces out of a deck of cards.
This is independent if you replace the card you drew, and dependent if you kept the card you drew.
Conditional probability is used when talking about something given that something else has happened. For instance, 70% of your friends like Chocolate,
and 35% like Chocolate AND like Strawberry. What percent of those who like Chocolate also like Strawberry?
Bayes Formula is used when we know other probabilities. For example, if the risk of developing health problems is known to increase with age,
Bayes' theorem allows the risk to an individual of a known age to be assessed
more accurately (by conditioning it on their age) than simply assuming that the individual is typical of the population as a whole.
Now let's look at how we can apply probability to some of the great paradoxes and problems of our time.
Betrand's Paradox is a fairly famous one. It states:
We are given three identical boxes, each with two drawers. Each drawer contains a coin. The first box has a gold coin in each drawer,
the second box has a gold coin in one drawer and a silver coin in the other, and the third box has a silver coin in each drawer. We choose
one of the boxes at random, then open one of its drawers, also at random. We discover a gold coin. What is the probability that the coin in
the other drawer of that box is also gold? (Flusser, 1984)
The Math:
They call this a paradox because at first glance you would say the answer is one-half. By discovering a gold coin, it must either
be from Box 1 or Box 2. If it is from Box 1 then the other coin will be gold, if not then it will be silver. Therefore there is an equally likely
chance of getting gold or silver so the probability is one-half. However, that is incorrect. By locating a gold coin first, it must either be from
Box 1 or Box 2. Box 1 and Box 2 have four total drawers, three have gold coins and one has a silver coin. Since a gold coin has been found,
that leaves three drawers with two gold and one silver coin. Therefore the probability of finding a gold coin in the next drawer is two-thirds.
Another famous probability problem is the birthday problem. The birthday problem asks,
What is the least number of people required to assure that the probability that two or
more of them have the same birthday exceeds one-half? (Mosteller, 1962)
The Math:
Many people are deceived by the question and believe the answer should be a lot higher, but that actually leads to a problem relating to birthmates.
The answer to this is 23. This answer is saying that in a room of 23 people the chance of two people having the same birthday is fifty percent.
One way to solve this is to look at the birthdays that are not matches. In a group of 23 people, there are 253 comparisons, or combinations,
that can be made. So, we're not looking at just one comparison, but at 253 comparisons. Every one of the 253 combinations has the same odds,
99.726027 percent, of not being a match. If you multiply 99.726027 itself 253 times, or calculate (364/365)253, you'll find there's a 49.952 percent
chance that all 253 comparisons contain no matches. Consequently, the odds that there is a birthday match in those 253 comparisons is 1-49.952
percent = 50.048 percent, or just over half. The more trials you run, the closer the actual probability should approach 50 percent (Buddies, 2012).
Buffon's Needle Problem has been around since the late 18th century. The problem states:
Suppose that you drop a short needle on ruled paper. What is then the probability that the needle comes to lie in a position
where it crosses one of the lines?
The Math:
Click Simulation to try the needle problem yourself.
The last famous probability problem that we will discuss is the Monty Hall Problem. This problem was made famous in the game show Let's Make A Deal.
The problem is as follows:
A contestant on a game show is given the choice of three doors to open. Behind one door is a prize; behind the other two doors is
nothing. The contestant picks a door, say No. 1, and the host, who knows what is behind the doors, opens another door, say No. 3, which has
nothing behind it. He then offers the contestant the option to switch to door No. 2. This is the problem: is it to the contestant's advantage to switch
the choice from door 1 to door 2? (Robinson, 2013)
The Math:
The answer to this problem is yes, it is to the contestant's advantage to switch doors. At the start of the game, each door is equally likely
to have the prize behind it. The probability of picking the correct door is one-third. The probability of picking an incorrect door is two-thirds.
The contestant chooses their door. They choose door 2. The host then opens door 3 because the host knows the prize is not behind that door.
Now there is only one door, its probability of having the prize is now two-thirds. This is why it is in the contestant's best interest to switch to
the other door.
