1. Monty Hall problem

2. 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. probabilityrandom variables  For discrete random variables, the joint probability mass function isdiscrete random variablesprobability mass function

3. 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. probability theorystatistics  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.random variablesjoint distributionprobability distributionjoint probability  For discrete random variables, the marginal probability mass function can be written as Pr(X = x).discrete random variablesmarginal probability

4. 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).discrete random variable  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.

5. Notations  For this problem the background is provided by the rules of the game and the propositions of interest are:  C i : The car is behind Door i, for i equal to 1, 2 or 3.  H ij: The host opens Door j after the player has picked Door i, for i and j equal  For example,  C 1 denotes the proposition the car is behind Door 1, and  H 12: denotes the proposition the host opens Door 2 after the player has picked Door 1.to 1, 2 or 3.

6. 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.a priori  In this context a priori means before the game is played, or before seeing the goat. Hence, the prior probability of a proposition C i is: P(C i )=1/3prior probability  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 H ij subject to where the car is conditional probability

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

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