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# Geometric distribution

### Question DP007

A rat has to choose between 5 doors, one of which contains chocolate.If the rat chooses the wrong door, it is returned to the starting point and chooses again, and continues until it gets the chocolate. Let X be the serial number of the trial on which the chocolate is found.

(a) Find the probability function of X

(b) What is the expectation of X?

Solution

(a) probability of getting chocolate on 1st attempt is p(X=1) = 1/5

probability of getting chocolate on 2nd attempt is p(X=2) = (4/5)(1/5)

probability of getting choco on rth attempt is p(X=r) = (4/5)^{r-1}(1/5)

Notice this pattern is the pdf of a geometric distribution with p =1/5.

(b) The expectation of a geometric distributed variable X is 1/p = 5.

Discuss the geometric distribution on the forum