We flip a FAIR coin until we get two heads in a row or two tails in a row. Let N

Question

We flip a FAIR coin until we get two heads in a row or two tails in a row. Let
N be the total number of flips and let X=N-1. Show that X is a geometric random
variable and find the value of the parameter p.
Remark. In this experiment the sample space consists of ALL infinite sequences
of tosses. N is the number of tosses required to obtain either two heads or two
tails, and we set it equal to if we never obtain either two consecutive heads
or two consecutive tails. The technical mathematical point is that the probability
of never obtaining either two consecutive heads or two consecutive tails is zero,
because such a sequence would be of the form (H, T, H, T, ...) or (T, H, T, H, ...).
By continuity of probability, both possibilities have probability limn(
1
2
)
n = 0.

Hence the possibility that N = can be ignored.

Explanation / Answer

so., here the following cases are possible

Case 1 +Case 2+---Case n

HH + TT + HTT + THH + HTHH + THTT +----

Now the cases are formed in such a fashion that first we can say min 2 chances and game ends and find the possibilities (i.e. HH & TT)

Now for 3 chances and game ends we have 2 possibilities (i.e. HTT & THH)

& so on

So., the probabilities associated with each case makes the total probabilities of 2 consecutive H ot T and making the game end

so.,

Total Probabilities = 0.5*0.5 + 0.5*0.5 + 0.5*0.5*0.5 + 0.5*0.5*0.5 + 0.5*0.5*0.5*0.5 + 0.5*0.5*0.5*0.5+---

= 2*(0.5*0.5 + 0.5*0.5*0.5 + 0.5*0.5*0.5*0.5+----)

=0.5 + 0.5*0.5 + 0.5*0.5*0.5 +---

the above series is Geometric progression series with r<1 and so to get the sum of it we just need to use the equation of sum of a GP = a/(1-r)

where a= first term =0.5

r = ratio of the 2nd and 1st term = 0.5*0.5/0.5 = 0.5

So., ans = a/(1-r) = 0.5/(1-0.5) =0.5/0.5 =1  

Hope the above answer has helped you in understanding the proble. Please upvote the ans if it has really helped you. Good Luck!!

Related Questions

Navigate

• Browse (All)
• Browse W
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.