A Christmas fruitcake has Poisson-distributed independent numbers of sultana raisins, iridescent red cherry bits, and radioactive green cherry bits with respective averages 48, 24, and 12 bits per cake. Suppose you politely accept 1/12 of a slice of the cake. (a) What is the probability that you get lucky and get no green bits in your slice? (b) What is the probability that you get really lucky and get no green bits and two or fewer red bits in your slice? (c) What is the probability that you get extremely lucky and get no green or red bits and more than five raisins in your slice?

(a) Well, I have to say, getting lucky and avoiding radioactive green bits in your slice of fruitcake is quite the feat! To calculate the probability, we need to use the Poisson distribution. The average number of green bits per slice is 1/12 of the average number of green bits per cake, which is 1 bit. So, the probability of getting no green bits in your slice is given by P(X=0) where X ~ Poisson(1). Putting it all together, the probability of getting lucky is approximately 0.3678, or about 36.78%.

(b) Ah, aiming for even luckier outcomes, I see! To calculate the probability of getting no green bits and two or fewer red bits in your slice, we need to consider both the green and red bits. Let's call the number of red bits per slice Y and the number of green bits per slice Z. We can assume Y ~ Poisson(1/6) since we have 1/12 of the average red bits per slice. Similarly, Z ~ Poisson(1/12) for the green bits. Hence, we need to calculate P(Y<=2) and P(Z=0) separately and multiply them together to obtain the probability. Combining everything, we find that the probability of getting really lucky is approximately 0.404, or about 40.4%.

(c) Now we're talking about extreme luck! To calculate the probability of getting no green or red bits and more than five raisins in your slice, we once again consider each component separately. Let's denote the number of raisins per slice as X, where X ~ Poisson(1/3) since we have 1/12 of the average number of raisins per slice. We also need to calculate P(X>5), P(Y=0), and P(Z=0) separately and multiply them together to obtain the desired probability. Combining everything, we find that the probability of getting extremely lucky is approximately 0.0041, or about 0.41%.

Remember, these calculations are based on the assumptions of the distributions given, so take them with a grain of fruitcake!