Probability of at least 4 consecutive heads occurring | 10 coin tosses | Exercise 1.1 (d)

When a fair coin is tossed 10 times, what is the probability of the event that at least 4 consecutive heads occur? Exercise 1.1d, Mitzenmacher and Upfal textbook "Probability and Computing" .
Let's solve together.

[@Zerker102]

great video but i have a question. how would one find the Probability of at least 4 consecutive heads occurring within 10 coin tosses, if it wasn't a fair coin toss? such as if the probability of heads and tails was 60% and 40%, respectively.

[@keithmasumoto9698]

I see. So the possibility of 5 or more consecutive heads is still in the calculation. The brilliance here is seeing all of these disjoint sets.

[@AK-dg4hk]

great word 👌 👏 sir ❤

[@learnwithsreyas1777]

Thank you!

[@parthapratimsamanta3982]

Very good explanation.
But is there any formula for this?
If yes then can you please share in the comment?

[@weinsterle1999]

P(L≥m) = Sum from k=1 to floor(n/m) [(-1)^(k+1)(p+(n-km+1)q/k)(n-km)C(k-1)p^(km)q^(k-1)]
where q=1-p and (n-km)C(k-1) is a binomial coefficient: "(n-km) choose (k-1)".
In the video we had m=4 ; n=10 ; p=q=1/2.
Do not ask me about the derivation because I have literally no idea. I only know that it works.

[@parthapratimsamanta3982]

@@weinsterle1999 Thank you so much for the reply.

[@theunbeatable340]

Where is 1024 comes frm?

[@learnwithsreyas1777]

Number of Total possible outcomes

[@dontaw5421]

@@learnwithsreyas1777 Do you have more lectures or solution from the same book?