Step 1 :Given that the number of trials (n) is 100 and the probability of success (p) is 0.63, we first need to check if the normal distribution can be used to approximate the binomial distribution. The rule of thumb is that the normal distribution can be used if both np and n(1-p) are greater than or equal to 5.
Step 2 :Calculate np and n(1-p): \(np = 100 * 0.63 = 63\) and \(n(1-p) = 100 * (1-0.63) = 37\). Both values are greater than 5, so we can use the normal distribution to approximate the binomial distribution.
Step 3 :Next, we need to find the probability that the number who have encountered fraudulent charges on their credit cards is at least 66. This can be done by calculating the z-score for 66 and finding the area under the normal distribution curve to the right of this z-score.
Step 4 :Calculate the mean and standard deviation of the distribution: mean = np = 63 and standard deviation = \(\sqrt{np(1-p)} = \sqrt{63 * 0.37} = 4.83\).
Step 5 :Calculate the z-score for 66: \(z = \frac{66 - mean}{std_dev} = \frac{66 - 63}{4.83} = 0.62\).
Step 6 :Finally, find the area under the normal distribution curve to the right of this z-score. This is the probability that the number who have encountered fraudulent charges on their credit cards is at least 66. The probability is 0.2672.
Step 7 :Final Answer: The probability that the number who have encountered fraudulent charges on their credit cards is at least 66 is \(\boxed{0.2672}\).