Monty Hall problem
Joint probability distribution
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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
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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.
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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).
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Marginal distribution
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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.
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Solution
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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
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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).
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
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For this problem the background is provided by the rules
of the game and the propositions of interest are:
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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,
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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
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The assumptions underlying the common interpretation of the
Monty Hall puzzle are then formally stated as follows.
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First the car can be behind any door, and all doors are a priori equally
likely to hide the car.
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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.
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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
=?