Some simple probability formulas with examples

A known relationship that is usually given axiomatically:

P(B|A) = \frac{{P(AB)}}{{P(A)}}

Upon rearrangement gives the multiplication rule of probability:

P(AB) = P(A)P(B|A) = P(B)P(A|B)

Now observe a cool set up that is handy to keep in mind for proving the law of total probability and Bayes’ theorem.

Imagine that B happens with one and only one of n mutually exclusive events A_1, A_2,..., A_n, i.e.:

 B = \sum\limits_{i = 1}^n {B{A_i}}

By addition rule:

B = \sum\limits_{i = 1}^n {P(B{A_i})}.

Now by multiplication rule:

B = \sum\limits_{i = 1}^n {P({A_i})P(B|{A_i})}.

This is the law of total probability

From the same set up imagine that we want to find the probability of even A_i if B is known to have happened. By the multiplication rule:

P(A_i B) = P(B)P(A_i|B) = P(A_i)P(B|A_i)

By neglecting P(A_i B) and dividing the rest through P(B) we get:

P\left( {{A_i}|B} \right){\rm{ = }}\frac{{P({A_i})P(B|{A_i})}}{{P(B)}}

And applying the law of total probability to the bottom we have the Bayes’ equation

P\left( {{A_i}|B} \right){\rm{ = }}\frac{{P({A_i})P(B|{A_i})}}{{\sum\limits_{j = 1}^n {P({A_j})P(B|{A_j})} }}

Bunch of examples:

Problem: P_t (k) is a known probability of receiving k phone calls during time interval t. Also k=0,1,2,.... Assuming that a number of received calls during two adjeicent time periods are independent find the probability of receiving s calls for the time interval that equal 2t.

Solution: Let A_{b.b + t}^k be an event consisted of k call in the interval b till b+t. Then clearly

A_{0,2t}^s = A_{0,t}^0A_{t,2t}^s + ... + A_{0,t}^sA_{t,2t}^0

which means that the event A_{0,2t}^s can be seen as sum of s+1 mutually exclusive events, such that in the first interval of duration t number of calls received is i and in the second interval of the same duration number of received calls is s-i (i=0,1,2,...,s). By rule of addition

P(A_{0,2t}^s) = \sum\limits_{i = 0}^s {P(A_{0,t}^iA_{t,2t}^{s - i})}.

By the rule of multiplication

P(A_{0,t}^iA_{t,2t}^{s - i}) = P(A_{0,t}^i)P(A_{t,2t}^{s - i})

If we change the notation so that

{P_{2t}}(s) = P(A_{0,2t}^s)

then

{P_{2t}}(s) = \sum\limits_{i = 0}^s {{P_t}(s) \cdot P(s - i)}.

It is known that under quite general conditions

{P_t}(k) = \frac{{{{(at)}^k}}}{{k!}}\exp \{ - at\} {\rm{ }}(k = 0,1,2...)

(Recall that the Poisson distribution is an appropriate model if the following assumptions are true. (a) k is the number of times an event occurs in an interval and k can take values 0,1,2,.... (b) The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently. (c) The rate at which events occur is constant. The rate cannot be higher in some intervals and lower in other intervals (that kinda a lot to take on faith really). (d) Two events cannot occur at exactly the same instant; instead, at each very small sub-interval exactly one event either occurs or does not occur. (e) The probability of an event in a small sub-interval is proportional to the length of the sub-interval.  Or instead of those assumptions, the actual probability distribution is given by a binomial distribution and the number of trials is sufficiently bigger than the number of successes one is asking about (binomial distribution approaches Poisson).)

Parametrisation then gives

{P_{2t}}(s) = \sum\limits_{i = 0}^s {\frac{{{{(at)}^s}}}{{i!(s - i)!}}\exp \{ - 2at\} } {\rm{ = }}{(at)^s}\exp \{ - 2at\} \sum\limits_{i = 0}^s {\frac{1}{{i!(s - i)!}}}

Note that

\sum\limits_{i = 0}^s {\frac{1}{{i!(s - i)!}} = \frac{1}{{s!}}\sum\limits_{i = 0}^s {\frac{{s!}}{{i!(s - i)!}}} = \frac{1}{{s!}}{{(1 + 1)}^s} = \frac{{{2^s}}}{{s!}}}

Then

{P_{2t}}(s) = \frac{{{{(2at)}^s}\exp \{ - 2at\} }}{{s!}}{\rm{ }}(s = 0,1,2,...)

The key point is that if for time interval t we have that parametrized formula for 2t we have the one above. It holds true for any multiples of t as well.

12 Replies to “Some simple probability formulas with examples”

  1. Have ʏou eνer thought about including a little Ьit
    more than ϳust youг articles? I mеаn, whuat уou sаy iѕ
    aluable and еverything. Howeνеr think about if yοu aԁded s᧐me gredat pictures or videos to giνe your
    posts more, “pop”! Your cointent is excellent
    but with images and clips, tһis website c᧐uld certаinly be
    one οf tһe very best іn its niche. Suprb blog!

  2. I believe that is one of the so much vital info for me.
    And i’m happy reading your article. However
    should statement on few general things, The site taste is ideal, the articles is really
    nice : D. Excellent task, cheers

  3. Woah! I’m really enjoying the template/theme of this site.
    It’s simple, yet effective. A lot of times it’s very
    hard to get that “perfect balance” between usability and appearance.
    I must say you’ve done a superb job with this. In addition, the
    blog loads super fast for me on Firefox. Superb Blog! It’s perfect time to make
    some plans for the future and it is time to be happy.
    I have read this post and if I could I want to suggest you few interesting things or tips.
    Perhaps you can write next articles referring to this
    article. I want to read even more things about it! Hey there would you mind stating which blog platform you’re using?
    I’m planning to start my own blog soon but I’m having a difficult time making a decision between BlogEngine/Wordpress/B2evolution and Drupal.
    The reason I ask is because your design seems
    different then most blogs and I’m looking for something unique.
    P.S My apologies for being off-topic but I had to ask!
    http://goodreads.com

  4. First of all I want to say wonderful blog! I had a quick question which I’d
    like to ask if you do not mind. I was interested to find out how you center yourself and clear
    your mind prior to writing. I’ve had trouble clearing my thoughts in getting my thoughts out there.
    I do take pleasure in writing but it just seems like the
    first 10 to 15 minutes are generally wasted simply
    just trying to figure out how to begin. Any recommendations or hints?
    Thank you! http://hatchsandwich.com/

  5. In this section we give most of the general derivative formulas and properties used when taking the derivative of a function. Examples in this section concentrate mostly on polynomials, roots and more generally variables raised to powers.

Leave a Reply

Your email address will not be published.