To calculate the standard deviation of number of recoveries among 48 patients when the probability of recovering is 0.75, we will use the formula for the standard deviation:

σ = √(npq)

Where:
n = sample size (48 patients)
p = probability of success (0.75)
q = probability of failure (1 - p = 0.25)



Before we can calculate the standard deviation, we need to find the mean of the number of recoveries. The mean is simply the expected value of the number of recoveries, which we can calculate using the formula:

E(X) = np

Where:
X = number of recoveries
n = sample size (48 patients)
p = probability of success (0.75)

E(X) = 48 * 0.75 = 36

Therefore, we expect 36 out of 48 patients to recover.



Now we can calculate the variance using the formula:

σ^2 = npq

σ^2 = 48 * 0.75 * 0.25 = 9

Therefore, the variance of the number of recoveries is 9.



Finally, we can calculate the standard deviation using the formula:

σ = √(npq)

σ = √(48 * 0.75 * 0.25) = √9 = 3

Therefore, the standard deviation of the number of recoveries is 3.



In conclusion, the standard deviation of the number of recoveries among 48 patients when the probability of recovering is 0.75 is 3. This means that we can expect the number of recoveries to be within 3 of the mean of 36.