Final answer:
The shape of the probability distribution of X, which follows a binomial distribution, is skewed to the right. The probability of testing 30 people and finding one with the disease is extremely low. The mean and standard deviation of the distribution can be calculated as np and sqrt(npq) respectively.
Step-by-step explanation:
The shape of the probability distribution of X, which follows a binomial distribution with parameters n=30 and p=0.59, will be skewed to the right. This means that the majority of the probabilities will be concentrated towards the left side of the distribution.
To sketch a graph of the distribution, you can plot the values of X on the x-axis, which range from 1 to 30, and their corresponding probabilities on the y-axis. The graph will show a decrease in probability as X increases, with a peak around the mean value.
To find the probability of testing 30 people to find one with the disease, you can calculate P(X=30) = (0.59)^1 * (0.41)^29, which is approximately 6.38692 x 10^(-16).
The probability of asking 10 people and finding one with the disease can be calculated as P(X=1) = (0.59)^1 * (0.41)^9 * (30 choose 1), which is approximately 0.14088.
The mean of the distribution can be calculated as np = 30 * 0.59 = 17.7, and the standard deviation can be calculated as sqrt(npq) = sqrt(17.7 * 0.41) = approximately 2.518.