9. Keebom rolls a biased die repeatedly until he observes that both an even number and an odd number appear. The probability that an even number will appear on a single roll is p, for 0

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9.
Keebom rolls a biased die repeatedly until he observes that both an even number and an odd
number appear. The probability that an even number will appear on a single roll is p, for
0 <p< 1.
Find the probability mass function of N, the number of rolls required to observe both an even
number and an odd number.
Hint: If N is the roll number that ends the experiment then that means that the N 1
rolls previous to roll N must all be the same as each other (either all even's or all odd's) but
different from the Nth roll. Also think about what the smallest value in the support of N
must be. Finally remember that there are two cases: a sequence of even's followed by an odd,
or a sequence of odd's followed by an even.)
Transcribed Image Text:9. Keebom rolls a biased die repeatedly until he observes that both an even number and an odd number appear. The probability that an even number will appear on a single roll is p, for 0 <p< 1. Find the probability mass function of N, the number of rolls required to observe both an even number and an odd number. Hint: If N is the roll number that ends the experiment then that means that the N 1 rolls previous to roll N must all be the same as each other (either all even's or all odd's) but different from the Nth roll. Also think about what the smallest value in the support of N must be. Finally remember that there are two cases: a sequence of even's followed by an odd, or a sequence of odd's followed by an even.)
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