Answer:
The probability that more than two-thirds of the tosses result in tails is approximately 0.0011.
Explanation:
To find the probability that more than two-thirds of the tosses result in tails, we need to calculate the probability of getting more than 60 tails out of 90 coin flips.
Since a fair coin has an equal probability of heads or tails, the probability of getting tails on any single flip is 0.5, and the probability of getting heads is also 0.5.
To calculate the probability of getting more than 60 tails, we can use the normal approximation to the binomial distribution when the number of trials is large (in this case, 90 coin flips).
First, we calculate the mean (μ) and standard deviation (σ) of the binomial distribution:
Mean (μ) = n * p = 90 * 0.5 = 45
Standard Deviation (σ) = sqrt(n * p * (1 - p)) = sqrt(90 * 0.5 * 0.5) = sqrt(22.5) ≈ 4.743
Next, we convert the problem into a normal distribution by applying the continuity correction. We subtract 0.5 from the threshold value (60 - 0.5) and use the z-score formula:
z = (x - μ) / σ
z = (60 - 0.5 - 45) / 4.743
z = 14.5 / 4.743
z ≈ 3.06
Now, we use the z-table or a calculator to find the probability associated with the z-score of 3.06. From the z-table, we find that the probability is approximately 0.0011.