On a TV news channel, the evening news starts at same time every day. The probability that Mr Li gets home from work in time to watch the news is $0.3$
In a particular week of five working days, what is the probability that Mr Li gets home in time to watch the news on three consecutive days?
Possible arrangements with three consecutive successes in 5 trials:
SSS.. FSSS. FFSSS SFSSS
I calculated the respective probabilities for these arrangements as follows:
$0.3^3 + 0.3^3 0.7 + 0.3^3 0.7^2+0.3^4 0.7 = 0.0648$
Actual answer (at the back of the book) = $0.05913$
My question is what I am doing wrong here (or is the book wrong?)
Also, I would really appreciate if someone could tell me a general way to solve the problems of this type where probability of $k$ consecutive success in $n$ trials is asked. In this question I was able to manually find the possible arrangements with consecutive successes but if the number of trials is high for example 200, how would I approach this problem then.
Thanks very much.