Question

a medical device company knows that the percentage of patients experiencing injection - site reactions with the current needle is 11%. what is the probability that an injection - site reaction occurs for the first time on the 6th patient of the day? 0.0001 0.0614 0.3685 0.4970

Explanation:

Step1: <Identify the probability model>

This is a geometric probability problem. The probability of success (injection - site reaction) \(p = 0.11\), and the probability of failure \(q=1 - p=1 - 0.11 = 0.89\). The formula for the geometric probability mass function is \(P(X = k)=q^{k - 1}p\), where \(X\) is the number of trials until the first success and \(k\) is the trial number.

Step2: <Substitute the values into the formula>

Here, \(k = 6\), \(p = 0.11\), and \(q = 0.89\). So \(P(X = 6)=(0.89)^{6 - 1}\times0.11\).

Step3: <Calculate \((0.89)^{5}\)>

\((0.89)^{5}=0.89\times0.89\times0.89\times0.89\times0.89\approx0.558\)

Step4: <Multiply by \(p\)>

\(P(X = 6)\approx0.558\times0.11 = 0.0614\)

Answer:

0.0614