Question

While investigating a group of defective solar arrays, taken from the Mohammed Bin Rashid Al Maktoum Solar Park in Dubai, you notice three main classifications of defects. 27% of defects are related to a cell’s semiconductor material bandgap, 51% are related to impact damage, and 22% are age-related in nature. Over the study period, 3 instances of material bandgap defects and 5 occurrences of age-related defects are recorded. What is the most probable number of recorded defects related to impact damage?

Note: This problem is related to discrete probability distribution ( I think multinomial distribution should be used)

Answer 1

Let p1, p2, p3 be the probabilities of defects due to material bandgap,  impact damage and age-related. Then,

p1 = 0.27, p2 = 0.51, p3 = 0.22

Let x be the number of recorded defects related to impact damage. Then by multinomial distribution, the probability that 3 instances of material bandgap defects and 5 occurrences of age-related defects and x occurrences of impact damage are recorded is,

P(x) = (3 + 5 + x)! / (3! * 5! * x!) * 0.27³ * 0.51^x * 0.22⁵

= (8 + x)! / (720 * x!) * 0.27³ * 0.51^x * 0.22⁵

P(x) ~ (8 + x)! * 0.51^x / (x!)

Thus, we need to find an integer value of x for which (8 + x)! * 0.51^x / (x!) is maximum

For x = 1, 2, ..., 10, we found (8 + x)! * 0.51^x / (x!) as

x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

(8 + x)! * 0.51^x / (x!) =  185068.8, 471925.4, 882500.6, 1350225.9, 1790399.5, 2130575.4, 2328414.6, 2374982.9, 2287900.2, 2100292.3

We see that the maximum value of (8 + x)! * 0.51^x / (x!) is 2374982.9 at x = 8

Thus,  most probable number of recorded defects related to impact damage is 8