3 different drugs are under consideration for trating a given ailment. in the past 10,000 applications, drug A was sed 2500 times, drug B 4000 times, and drug C 3500 times. when an expert was asked to rank the effectiveness of the 3 drugs, he replied: "drugs A is twice as likely to be used as drug B, and drug, B is 3 times as likely to be used as drug C". what is the probability of using each drug in the nest application, according to the:
a) classical theory
B)relative frequency theory
c)personalistic theory
a) According to the classical theory, the probability of an event is determined by counting the number of favorable outcomes and dividing it by the total number of possible outcomes. In this case, there are 3 drugs, so the total number of possible outcomes is 3.
Given the expert's statement, drug A is twice as likely to be used as drug B, and drug B is three times as likely to be used as drug C. We can assign probabilities as follows:
Let P(A) represent the probability of using drug A.
Let P(B) represent the probability of using drug B.
Let P(C) represent the probability of using drug C.
Based on the expert's statement:
P(A) = 2P(B)
P(B) = 3P(C)
To find the probabilities, we need to set up a system of equations using the fact that the sum of probabilities of all possible outcomes is 1:
P(A) + P(B) + P(C) = 1
Substituting the values from the expert's statement, we get:
2P(B) + P(B) + P(C) = 1
3P(B) + P(C) = 1
We also know that P(A) + P(B) + P(C) = 1, so we can substitute P(A) and P(B) using the relationships given by the expert:
2P(B) + P(B) + P(C) = P(A) + P(B) + P(C)
2P(B) + P(B) + P(C) = 2P(B) + 3P(C) + P(C)
2P(B) + P(B) + P(C) = 2P(B) + 4P(C)
Simplifying the equation:
P(B) = 3P(C)
Substituting this back into the earlier equation:
3P(B) + P(C) = 1
3(3P(C)) + P(C) = 1
9P(C) + P(C) = 1
10P(C) = 1
P(C) = 1/10
Substituting this value back into P(B):
P(B) = 3P(C) = 3(1/10) = 3/10
Finally, substituting P(C) and P(B) into P(A):
P(A) = 2P(B) = 2(3/10) = 6/10 = 3/5
Therefore, according to the classical theory, the probability of using drug A in the next application is 3/5, the probability of using drug B is 3/10, and the probability of using drug C is 1/10.
b) According to the relative frequency theory, the probability of an event is determined by calculating its relative frequency based on past observations. In this case, the relative frequencies can be calculated as:
Relative Frequency of Drug A = 2500/10000 = 0.25
Relative Frequency of Drug B = 4000/10000 = 0.4
Relative Frequency of Drug C = 3500/10000 = 0.35
Therefore, according to the relative frequency theory, the probability of using drug A in the next application is 0.25, the probability of using drug B is 0.4, and the probability of using drug C is 0.35.
c) The personalistic theory refers to subjective probabilities assigned by an individual based on personal beliefs, experiences, or opinions. Since no specific information is given about the expert's personal beliefs or opinions, we cannot determine the probabilities according to the personalistic theory. It would require further input or clarification from the expert.