The Happy Animals Kennel has $18$ cages in a row. They allocate these cages at random to $6$ dogs, $6$ cats, and $6$ pot-bellied pigs (with one animal per cage). All arrangements are equally likely. Let $A$ be the number of times in the row of cages that two animals of the same species are adjacent. For example, in the arrangement DCPCDPPPDCDPCDCCPD (where D=dog, C=cat, and P=pig), we have $A=3$. What is the expected value of $A$?

Which best describes the strength of the correlation, and what is true about the causation between the variables? it is a weak negative correlation, and it is not likely causal. it is a weak negative correlation, and it is likely causal. it is a strong negative correlation, and it is not likely causal. it is a strong negative correlation, and it is likely causal.

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Amachine that produces a special type of transistor (a component of computers) has a 2% defective rate. the production is considered a random process where each transistor is independent of the others. (a) what is the probability that the 10th transistor produced is the first with a defect? (b) what is the probability that the machine produces no defective transistors in a batch of 100? (c) on average, how many transistors would you expect to be produced before the first with a defect? what is the standard deviation? (d) another machine that also produces transistors has a 5% defective rate where each transistor is produced independent of the others. on average how many transistors would you expect to be produced with this machine before the first with a defect? what is the standard deviation? (e) based on your answers to parts (c) and (d), how does increasing the probability of an event a↵ect the mean and standard deviation of the wait time until success?