Final answer:
To find the probability of no counts in a 15-second interval for a Geiger counter that has an average of three counts per minute, we apply the Poisson probability formula. The average rate for this interval is 0.75 counts, and the probability of observing no counts is e^-0.75, which is approximately 0.4724 or 47.24%.
Step-by-step explanation:
The Geiger counter counts of a radioactive sample following a Poisson process with an average rate is given as three counts per minute. To find the probability of no counts in a 15-second interval, we utilize the formula for the Poisson probability: P(X = k) = (λ^k * e^−λ) / k!, where X is the random variable representing the number of events (or counts), k is the number of events we are trying to find the probability for, λ is the average rate, and e is the base of the natural logarithm.
In a 15-second interval, since there are 3 counts per minute on average, the rate λ for our interval is (3 counts/minute) * (1 minute/4) = 0.75 counts per 15 seconds (since one minute has 60 seconds, and our interval is 15 seconds, therefore it's 1/4 of the minute). The probability of observing no counts (k = 0) in that interval is given by: P(X = 0) = (0.75^0 * e^−0.75) / 0!
Since 0! is 1, this simplifies to P(X = 0) = e^−0.75. Substituting the value of e (approximately 2.71828), we calculate: P(X = 0) ≈ e^−0.75 ≈ 0.4724. Therefore, the probability that there are no counts in a 15-second interval is approximately 0.4724, or 47.24%.