Talk:High School Mathematics Extensions/Discrete Probability

From Wikibooks, the open-content textbooks collection

Ok, to tell you the truth: I do not know much about discrete probabilities. I'm very interested to see how it will develop. I've just seen a A.Level maths exam paper, and the first question is about probability.

P(A)=0.8,P(B)=0.75,P(A"intersection"B)=k.

a)Express P(A"Union"B) in terms of k.

(My)Answer:We sum up P(A) and P(B), but then the probabilities that both A and B are true are counted twice, therefore we subtract k. So it is 0.8+0.75-k=1.55-k

b)Hence show that 0.55≤k≤1

(My)Answer:Since all probabilities are smaller than or equal to 1, k must be at least 0.55. Since k itself is also a probability, it must be smaller than or equal to 1. Putting these together, we get 0.55≤k≤1

...

--Lemontea 11:49, 3 Feb 2005 (UTC)

All correct, and explained very well. To bring your maths to uni level (which you are more than capable of) you need to quote the De Morgan's laws for part a) and actually solve 0 <= 1.55 - k <= 1 algebraicly and use the fact that 0 <= k <=1, for part b). 10/10. Xiaodai 00:25, 6 Feb 2005 (UTC)

I've added some information on adding and multiplying probabilities -- I hope it's all okay. Comments and revisions, of course, are welcome. On another note, I wanted to ask if anyone knows what the final section on options pricing has to do with discrete probability. I personally can't find anything relevant in it.

--Buggi22 07:08, 26 May 2005 (UTC)

That's okay, and it's an essential part of the topic. However, may I point out that the multiplacation law of probability does not always apply: It is true only when they are independent events. But then, how will we introduce the concept of dependent and independent events, conditional probability etc? Should we put them before or after your section. I'm at a lost at this moment and are looking forward for your work. --Lemontea 09:42, 26 May 2005 (UTC)

Option pricing is an important part of finance mathematics. Although it doesn't really use probability in the true sense, it does contain some material on "arbitrage free" probabilities which i think is interesting. Xiaodai 11:08, 27 May 2005 (UTC)

Contents 1 Tree diagram 2 Misconception 3 probability space section feels clunky 3.1 Definition -- Probability Space 3.2 Exercises

[edit] Tree diagram

why there are no three diagrams? 06:14, 22 October 2009 (UTC)~~

[edit] Misconception

I believe there is bias here. I am not expert on the field so I don't dare to edit the page directly. I am talking about this part:

Please note that the probability 1/6 does not mean that it will turn up 1 in at most six tries. Its precise meaning will be discussed later on in the chapter. Roughly, it just means that on the long run (i.e. the die being tossed a large number of times), the proportion of 1's will be very close to 1/6.

There are two school of thought that interpret what does probability means:

http://en.wikipedia.org/wiki/Probability#Interpretations

That misconception just reflect the 1st school of thought, the frequentist while Bayesian school of thought is not properly represented here. 05:56, 22 October 2009 (UTC)05:56, 22 October 2009 (UTC)ArielGenesis (talk) 05:56, 22 October 2009 (UTC)

[edit] probability space section feels clunky

I have pasted the following thing here so we can re-use the material later if need be:

We have now all the machinery required to define a Probability Space. Before we do that, let's consider the example of throwing a die again. If we throw a die, the numbers that will turn up are 1, 2, 3, 4, 5 or 6, and nothing else. Notice the events are disjoint and their probabilities add up to 1, i.e. all the possible events have been accounted for.

[edit] Definition -- Probability Space

A finite probability space of a random experiment consists of events:

E₁,E₂,....E_n

such that:

1. P(E₁) + P(E₂) + ... + P(E_n) = 1
2. all events are disjoint

In the above definition a random experiment is simply an action like tossing a coin or picking a card, where the outcome is uncertain. For example, let the random experiment be tossing a die, and let the events be the die turning up 1, 2, 3, 4, 5, or 6. So there are 6 events, and they are all disjoint, and the probabilities of each of the events add up to 1, i.e. all the possible events are included. Therefore it's a probability space.

Example 1

Let the random experiment be choosing a card randomly from a complete deck. And let the events be the card is of the suit Spades, Hearts, Clubs or Diamonds. Is this a probability space?

Solution To check whether it is a probability space, we need to check two conditions.

1. There are 4 events, and their probabilities add up to 1.
2. Each of the events are disjoint

so it's a probability space.

Example 2

Let the random experiment be tossing a coin twice. Let the events be

the first toss results in a head, or

the second toss results in a tail

Is this a probability space?

Solution We proceed to check the two conditions:

1. There are 2 events. Each event has probability of 1/2 (check) and so they add up to 1.
2. The events are not disjoint! So it is NOT a probability space.

[edit] Exercises

1. Let the random experiment be tossing a coin once. Let the 2 events be tossing H or T. Is this a probability space

2. Let the random experiment be tossing a die three time. Let the 3 events be the maximum of the 3 die throws are

1. 1 or 2,
2. 3 or 4, or
3. 5 or 6

. Is this a probability space

3. Give an exmaple of an experiment that satisfies the first condition but not the second.

4. Give an exmaple of an experiment that satisfies the second condition but not the first.

5. Let S be a probability space. It has n events

E₁,E₂,...E_n

Use the Simple Inclusion Exclusion Principle, show that

for i ≠ j.

6. Is it true that the complement of any event E_i is the union of all the other events i.e. is the following true:

(Hint: use )--Xiaodai 05:57, 6 August 2005 (UTC)