Question

Consider again the distribution of number of centipedes per square from the previous problem. Let p = 0.55 be the probability of at least one centipede in a square. Now suppose a total of n = 10 squares are randomly selected from the beech woods. Let Y be the number of squares out of the ten that have at least one centipede. So Y~binomial (10,0.55).

(a) What is the mean of Y?
(b) What is the probability that there are no centipedes at all in the ten sampled squares?
Show by hand and using R code.
(c) What is the probability that centipedes are found in at least one of the ten squares?

Answer 1

The mean of Y is 5.5. The probability of no centipedes in the ten sampled squares is 0.000548. The probability of centipedes being found in at least one of the ten squares is 0.999452.

Step-by-step explanation:

(a) To find the mean of Y, we can multiply the number of trials (n) by the probability of success (p). The mean of a binomial distribution is given by μ = np. In this case, n = 10 and p = 0.55. So the mean of Y is μ = 10 * 0.55 = 5.5.

(b) To find the probability of no centipedes in the ten sampled squares, we can use the binomial probability formula: P(X = 0) = (1 - p)^n. In this case, p = 0.55 and n = 10. So P(X = 0) = (1 - 0.55)^10 = 0.000548 (rounded to six decimal places).

(c) To find the probability of centipedes being found in at least one of the ten squares, we can use the complement rule: P(Y ≥ 1) = 1 - P(Y = 0). From part (b), we know that P(Y = 0) = 0.000548. So P(Y ≥ 1) = 1 - 0.000548 = 0.999452 (rounded to six decimal places).