Question

The backoff torque required to remove bolts in a steel plate is rated as high, moderate, or low. Historically, the probability of a high, moderate, or low rating is 0.68, 0.25, or 0.07, respectively. A sample of 16 bolts are selected for testing. Let x, y, and z denote the number of bolts that are independently rated as high, moderate, and low, respectively. Determine the following probabilities. (a) P(x = 11, y = 2, z = 3) (b) P(x = 9 | y = 6)

Answer 1

The probability of getting specific combinations of rated bolts among 16 tested can be determined using the multinomial and binomial distributions, based on given probabilities for each rating. The probabilities are calculated following their respective formulas considering the independence of each bolt's rating.

Step-by-step explanation:

The probabilities for the backoff torque ratings to remove bolts in a steel plate are 0.68 for high, 0.25 for moderate, and 0.07 for low. Using these probabilities, we can determine the probability of specific outcomes when sampling 16 bolts. The ratings of each bolt are independent of each other, suggesting the use of the multinomial distribution to calculate the probabilities in question.

(a) To determine P(x = 11, y = 2, z = 3), since there are 16 bolts tested (x + y + z = 16), we can use the formula for multinomial probabilities:

P(x = 11, y = 2, z = 3) =
16! / (11! × 2! × 3!) × (0.68^11) × (0.25^2) × (0.07^3)

(b) To determine P(x = 9 | y = 6), we need to calculate the conditional probability assuming that there are 6 moderate bolts (y = 6). Since y is already given, the sample size for x and z is reduced to 10 (16 - 6). The probability of getting exactly 9 high ratings out of these 10 bolts is calculated using the binomial distribution:

P(x = 9 | y = 6) = 10! / (9! × 1!) × (0.68^9) × (0.32^1)

The actual calculations of these probabilities will involve using a calculator to compute the factorials and powers involved in the equations above.