 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/discrete-mathematics-continuous-mathematics-and-probability-resit/7/b/ii/solution

Write

G (s) = E [s^{F}]

. The initial possibilities are

H

1/2

, after which a fresh wait begins;

T H

1/4

, after which a fresh wait begins; or

TT

1/4

, which ends the experiment. Accounting for the flips already made in each case gives the probability generating function equation

G (s) = \frac{s}{2} G (s) + \frac{s ^{2}}{4} G (s) + \frac{s ^{2}}{4} = \frac{( 2 s + s ^{2} ) G ( s ) + s ^{2}}{4} .

Therefore $G_{F} (s) = \frac{s ^{2}}{4 - 2 s - s ^{2}}$ for

0 \leq s \leq 1

. In particular,

G_{F} (1) = 1

and there is no mass below two flips, as required.

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