91Ó°ÊÓ

In the context of probability and statistics, the "Probability of Success" refers to the likelihood of a specific outcome occurring in an experiment or trial. Here, for the transducer calibration to conform to specifications, the probability of success is given as 0.6 or 60%. This means that on any given attempt, there is a 60% chance that the calibration will meet the required specifications.
Understanding the probability of success is crucial as it sets the foundation for calculating the likelihood of different outcomes over multiple attempts. In our exercise, knowing the probability of success lets us use the geometric distribution to analyze how many calibration attempts might be needed for the first success.
This concept is also valuable because it helps to determine the parameters needed for further probability calculations. For example, with a probability of success of 0.6, the probability of failure on a single attempt is the complement, which is 0.4 or 40%.

The term "Independence of Events" in probability refers to a scenario where the outcome of one event does not affect the outcome of another. In our exercise, each calibration attempt is considered an independent event. This means the result of one calibration has no bearing on the next.
Because of this independence, the assumptions of a geometric distribution are satisfied. Specifically, this distribution is concerned with the probability of the first success in a series of independent and identically distributed Bernoulli trials, where each trial results in success or failure.
In practical terms, for our situation, independence means that no matter how many times we have failed or succeeded in previous calibration attempts, the probability of success on the next attempt remains at 0.6. This forms the core principle for calculating further probabilities involving the number of attempts needed to achieve a calibration that conforms to specifications.

To calculate the probability of achieving success within a certain number of attempts, we use the concept of probability calculation within the geometric distribution. For our exercise, we are interested in finding the probability that success occurs on or before the third attempt.
The formula to calculate this in a geometric distribution where the success probability is given as \( p = 0.6 \) is \( P(X = k) = (1-p)^{k-1} imes p \). Here, \( (1-p) \) represents the probability of failure. This formula helps us determine the probability that the first success occurs exactly on the \( k \)-th attempt.
Thus, for each attempt:

• First attempt: \( P(X=1) = p = 0.6 \).
• Second attempt: \( P(X=2) = (1-p) \times p = 0.4 \times 0.6 = 0.24 \).
• Third attempt: \( P(X=3) = (1-p)^2 \times p = 0.4^2 \times 0.6 = 0.096 \).

To find the probability of success on or before the third attempt, sum these probabilities: \[ P(X \leq 3) = 0.6 + 0.24 + 0.096 = 0.936 \]. This result means there is a 93.6% chance of success within three calibration attempts.