Hi, how do I prove that the sum of all the probabilities is equal to 1 for this problem?
Flip a coin until heads show, assume that the probability of heads on one flip is 4/5. We define a random variable X = the number of flips.
a) What are the possible values of X?
b) Find the probability distribution for X: give the first four values and then find a general formula for the probability that X = n
c) Prove that the sum of all probabilities is 1 using the formula for the sum of a geometric series.
p(1) = 4/5
p(2) = 4/25
p(3) = 4/125
p(n) = ((1/5)^n-1) * (4/5)
The series could continue for ever so you need the sum to infinity of a geometric series.
Here 'a' is the first term 4/5 and r is 1/5
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