Final answer:
The probability that the pain will be relieved in less than 6 minutes is approximately 0.421, calculated using the exponential distribution with the rate parameter as the reciprocal of the mean time for pain relief.
Step-by-step explanation:
The student has asked to find the probability that the pain is relieved in less than 6 minutes given that the time it takes for an aspirin to relieve pain is exponentially distributed with an average time of 11 minutes. To solve this, we use the formula for the exponential distribution probability density function:
P(X < x) = 1 - e^{-(λx)}
Where λ (lambda) is the rate parameter, which is the reciprocal of the mean (λ = 1/11). To find the probability that the pain is relieved in less than 6 minutes, we calculate:
P(X < 6) = 1 - e^{-(1/11) ∙ 6}
Calculating the exponent:
e^{-(1/11) ∙ 6} ≈ e^{-0.5454545...}
Which is roughly e^{-0.5455} and the value of e^-0.5455 is approximately 0.5793. Hence, the probability is:
P(X < 6) = 1 - 0.5793
Pain relief probability: ≈ 0.421