Question

Suppose we toss a coin 20 times in a row. We are considering how many times it happens that we have an outcome `heads' and the next outcome is `tails'. This is a random variable since it depends on outcome of our tosses. What is the expectation of this random variable?

Answer 1

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Answer 2

Answer:

probability of getting a head or tail is 1/2.

In 20 tossing, the probability of a particular sequence of head/tails is 1/2^20

if we fix HT as one slot. there can be 19 such slots. in remaining 18 slots, there can be from no heads to all 18 heads. no. of possible arrangements will be as given below

all tail+ one head + two head +.....+ all heads

18C0 + 18C1 + 18C2 + 18C3+........+18C18

(1+1)^18

= 2^18

hence total number of ways of getting HT is

19*2^18

hence the expectation is

19*2^18/2^20

19/4