Question

let the probability of a patient recovering from a psychological disorder after a treatment is
0.75. Suppose there are 4 patients. X denotes number of patients recovered. Find the
probability of P(X=2)​

Answer 1

P(X=2) = 0.2109375.

Given,

The probability of a patient recovering from a psychological disorder after a treatment is

0.75

To find,

The probability of P(X=2), when there are 4 patients.

X denotes number of patients recovered.

Solution:

The probability of a patient recovering from a psychological disorder after a treatment,

• p = 0.75

The probability of a patient not recovering from a psychological disorder after a treatment,

• q = 1 - p
• q = 1 - 0.75
• q = 0.25

Total number of patients = 4,it gives

• P(X=2) = ⁴C2 * p²* q²
• P(X=2) = ⁴C2 * (0.75)²* (0.25)²
• P(X=2) = 4!/(2!*2!) * (0.75)²* (0.25)²
• P(X=2) = 6 * (0.75)²* (0.25)²
• P(X=2) = 0.2109375

Hence, P(X=2) = 0.2109375.

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Answer 2

Answer:

Hence, the probability P (X=2) = 0.21093.

Explanation:

Let the probability of recover be p = 75%= 0.75

Probability of non recovery (death) q = 25% = 0.25

Total number of patients = 4

The probability follows a binomial distribution with n = 4 and p = 0.75

P(X=2) = $^{4} C_{2}$* p²* q²

P(X=2) = $^{4} C_{2}$* (0.75)²* (0.25)²

P(X=2) = $\frac{4!}{2! \times2!}$ * (0.75)²* (0.25)²

P(X=2) = 6 * (0.75)²* (0.25)²

P(X=2) = 0.2109375

Therefore, P(X=2) = 0.21093.

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