Questions: A psychologist determined that the number of sessions required to obtain the trust of a new patient is either 1,2 , or 3. Let x be a random variable indicating the number of sessions required to gain the patient's trust. The following probability function has been proposed. f(x) = x / 6 for x=1,2, or 3 a. Consider the required conditions for a discrete probability function, shown below. f(x) ≥ 0 sum f(x) = 1 Does this probability distribution satisfy equation ( 5.1 )? Yes, all probability function values are greater than or equal to 0 Does this probability distribution satisfy equation (5.2)? Yes. the sum of all probability function values equals 1 b. What is the probability that it takes exactly 2 sessions to gain the patient's trust (to 3 decimals)? 1 / 3 c. What is the probability that it takes at least 2 sessions to gain the patient's trust (to 3 decimals)?

Solution Steps

To solve the given problem, we need to verify the conditions for a discrete probability function and calculate specific probabilities based on the provided probability function.

a. We need to check if the probability function values are non-negative and if their sum equals 1.

b. We need to find the probability that exactly 2 sessions are required, which is given by the probability function for \(x = 2\).

c. We need to calculate the probability that it takes at least 2 sessions, which is the sum of the probabilities for \(x = 2\) and \(x = 3\).

To determine if the given probability function \( f(x) = \frac{x}{6} \) for \( x = 1, 2, 3 \) satisfies the conditions of a discrete probability distribution, we check:

1. Non-negativity: \[ f(1) = \frac{1}{6} \geq 0, \quad f(2) = \frac{2}{6} = \frac{1}{3} \geq 0, \quad f(3) = \frac{3}{6} = \frac{1}{2} \geq 0 \] Thus, \( f(x) \geq 0 \) for all \( x \).

2. Sum of Probabilities: \[ \sum_{x=1}^{3} f(x) = f(1) + f(2) + f(3) = \frac{1}{6} + \frac{1}{3} + \frac{1}{2} = \frac{1}{6} + \frac{2}{6} + \frac{3}{6} = \frac{6}{6} = 1 \] Therefore, \( \sum f(x) = 1 \).

Both conditions are satisfied.

To find the probability that it takes exactly 2 sessions to gain the patient's trust, we evaluate: \[ P(X = 2) = f(2) = \frac{2}{6} = \frac{1}{3} \]

To find the probability that it takes at least 2 sessions, we calculate: \[ P(X \geq 2) = P(X = 2) + P(X = 3) = f(2) + f(3) = \frac{2}{6} + \frac{3}{6} = \frac{5}{6} \]

Final Answer