Monty Hall problem

Joint probability distribution In the study of probability, given two random variables

 X and Y, the joint distribution of X and Y defines the probability of events defined in terms of both X and Y. 

For discrete random variables, the joint probability mass function is

Marginal distribution In probability theory and statistics, the marginal

distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. 

Given two random variables X and Y whose joint distribution is known, the marginal distribution of X is simply the probability distribution of X averaging over information about Y. This is typically calculated by summing or integrating the joint probability distribution over Y.

For discrete random variables, the marginal probability mass function can be written as Pr(X = x).

Marginal distribution Imagine for example you want to compute the probability that a

pedestrian will be hit by a car while crossing the road at a pedestrian crossing. Let H be a discrete random variable describing the probability of being hit by

a car while walking over the crossing, taking one value of (hit, not hit). Let L be a discrete random variable describing the probability of the light

being in any particular state at a given moment, taking one of (red, yellow, green).

Solution P(L=red) = 0.6, P(L=yellow) = 0.1, P(L=green) = 0.3 P(H=hit|L=red) = 0.01, P(H=hit|L=yellow) = 0.09, P(H=hit|L=green) = 0.9 P(H=hit) = 0.6*0.01 + 0.1*0.09 + 0.3*0.9 = 0.285

It is important to interpret these results correctly even though these figures are contrived and the likelihood of being hit while

crossing at a red light is probably a lot less than 1%, the chance of being hit by a car when you cross the road is obviously a lot less than 28.5%.

if you were to put on a blindfold, wear earplugs, and cross the road at some random time, you'd have a 28.5% chance of being hit by a car, which seems more reasonable.

Notations For this problem the background is provided

by the rules of the game and the propositions of interest are: Ci : The car is behind Door i, for i equal to 1, 2 or

3.  Hij: The host opens Door j after the player has

picked Door i, for i and j equal For example,  

C1denotes the proposition the car is behind Door 1, and  

H12: denotes the proposition the host opens Door 2 after the player has picked Door 1.to 1, 2 or 3.

Assumptions The assumptions underlying the common interpretation of

the Monty Hall puzzle are then formally stated as follows. First the car can be behind any door, and all doors are a priori

 equally likely to hide the car. In this context a priori means before the game is played, or before seeing

the goat. Hence, the prior probability of a proposition  Ci  is: P(Ci )=1/3

Second, the host will always open a door that has no car behind it, chosen from among the two not picked by the player. If two such doors are available, each one is equally likely to be opened.  This rule determines the conditional probability of a proposition Hij subject

to where the car is 

Player picks Door 1, host opens Door 3 (not switch door)

Player picks Door 1, host opens Door 3 (switch door)- witching to Door 2

=?