 Hello and welcome to the session. I am Deepika here. Let's discuss the question which says How many times must have been toss a fair coin so that the probability of having at least one half is more than 90 percent Now we know that probability of X successes is equal to N C X Into Q raised to power N minus X into K raised to power X where X is from 0 to N and Q is equal to 1 minus P So this is a key idea behind our question We will take the help of this key idea to solve the above question So let's start the solution now according to the given question How many times must a man toss a fair coin so that The probability of having at least one hat is more than 90 percent So let the man tosses a fair coin N times we know that the repeated tosses of a coin are Bernoulli trials so each trial we have the probability of getting a hat that is P is equal to 1 over 2 and the probability of not getting a hat that is the probability of getting a tail That is Q is equal to 1 over 2 now we have According to our key idea probability of X successes is equal to NC X into Q raised to power N minus X into P raised to power X So this is equal to NC X Into 1 over 2 raised to power N minus X into 1 over 2 raised to power X Which is again equal to NC X into 1 over 2 raised to power N Now we are given that the probability of having at least one hat is more than 90 percent so According to the given question we have probability of getting at least one head is greater than 90 percent That is probability of X greater than equal to 1 is greater than 90 over 100 therefore 1 minus probability of X equal to 0 is greater than 0.9 or 1 minus NC 0 Into 1 over 2 raised to power N is greater than 0.9 or NC 0 into 1 over 2 raised to power N is less than 1 minus 0.9 that is 1 over 2 raised to power N is less than 0.1 or 2 raised to power N is greater than 1 over 0.1 or 2 raised to power N is greater than 10 This implies is greater than equal to 4 hence the man must toss a fair coin four times Hence the answer for top of question is N is greater than equal to 4 So this completes our session. I hope the solution is clear to you. Bye and have a nice day