Question

A fair coin is tossed until a head or five tails occurs. find the expected number of tosses of the coin.

Answer 1

Expected number of tosses = 20

The tossing of going will stop when either a head or five tail occurs.

Possible tosses -

When head occurs in the first toss, when head occurs in second toss and so on till head occurs in the fifth toss .

During the fifth toss if tail occurs and head has not occured , the coin tossing will stop.

Total tosses = 1 (H) +2 (T,H)+3(T,T,H)+ 4(T,T,T,H) +5(T,T,T,T,H) + 5(T,T,T,T,T)

= 20

Answer 2

Answer:

1.9

Step-by-step explanation:

The sample space related to the given random experiment is as follows:

S = {H,TH,TTH,TTTH,TTTTH,TTTTT}

It is clear that X assumes values 1 ,2 ,3 ,4 , 5 such that,

P (X = 1) = P(H) = 12  

P(X = 2) = P(TH) = P(T) P(H) = 14  

P(X = 3) = P(TTH) = P(T)P(T)P(H) = 12×12×12 = 18  

P(X = 4) = P(TTTH) = P(T) P(T) P(T) P(H) = 12×12×12×12=116,

and, P (X = 5) = P (TTTTH ∪ TTTTT)

⇒ P(X = 5) = P(TTTTH) + P(TTTTT)

⇒ P (X = 5) = P (T) P (T) P (P) P (T) P (H) + P(T) P(T) P(T) P(T) P(T) = (12)5+(12)5=116  

Therefore, the probability distribution of X is as follows:  

xi  1 2 3 4 5

pi  12 14 18 116 116

∴ Mean = Σpixi = 12×1+14×2+18×3+116×4+116×5 = 3116 = 1.9