A fair coin is tossed 10001000 times. Use CLT to compute the probability that head appears at most 540540 times. Enter the answer correct to 2 decimal places.

The Central Limit Theorem (CLT) states that the sum of a large number of independent and identically distributed random variables will be approximately normally distributed.

In this case, we have a fair coin, so the probability of getting a head (p) is 0.5. The number of trials (n) is 1000.

1. First, we calculate the mean (μ) and the standard deviation (σ). For a binomial distribution, the mean is np and the standard deviation is sqrt(np(1-p)).

   μ = np = 1000 * 0.5 = 500 σ = sqrt(np(1-p)) = sqrt(1000 * 0.5 * 0.5) = 15.81

2. Next, we standardize the score of 540. We subtract the mean from the score and divide by the standard deviation. This gives us the z-score, which tells us how many standard deviations above or below the mean the score is.

   z = (540 - 500) / 15.81 = 2.53

3. Finally, we look up the z-score in a standard normal distribution table or use a calculator to find the probability. The table or calculator gives the probability that a score is less than the z-score.

   P(Z < 2.53) = 0.9943

So, the probability that a head appears at most 540 times is approximately 0.9943 or 99.43% when rounded to two decimal places.