Question

2. In the Gulf of Maine, 7different species (two types of mussels, barnacles, and four types of seaweed) colonize the rocky intertidal zone between the low-and high-tide marks. These species are sessile, meaning they attach to the rock, so they are in competition with each other for space.

Within any square meter, there are always 4 and only 4 species. How many possible ways can a square meter be inhabited?

Within any square meter, the spaces occupied by the species are never equal. You realize that the colonization sequence affects how much spacea species can occupy. How many possible ways can a square be colonized?

3. You are a climate scientist interested in lightning strikes. You install a lightning rod on each of the 6 highest mountain peaks surrounding a valley. During a storm, the probability of lightning striking a rod is 0.2.

Without using a table, what is the probability that that none of the rods will be struck?

Using a table, calculate the same probability.

To create a fantastic meterological phenomenon, all 6 rods must be struck. What is the probability of this?

Answer 1

Solution

Back-up Theory

Number of ways of selecting r things out of n things is given by ⁿC_r = (n!)/{(r!)(n - r)!}……(1)

Values of ⁿC_r can be directly obtained using Excel Function: Math & Trig COMBIN……. (1a)

Now to work out the solution,

Part (1)

Given 7 different species and ‘Within any square meter, there are always 4 and only 4 species.’ Number of possible ways a square meter can be inhabited.’ Vide (1), is

⁷C₄ = 35   Answer 1

Part (2)

Let 7 species in the order of their sizes (assuming the space occupied is proportional to the size) be: S1, S2, S3, S4, S5, S6, S7.

Then colonization sequences can be:

S1, S2, S3, S4;   S2, S3, S4, S5; S3, S4, S5, S6; S4, S5, S6, S7.

Thus, the answer is: 4 Answer 2

Part (3)

Required probability = (1 – 0.2)⁶Answer 3

= 0.8⁶

= 0.2621 Answer 4

Part (4)

Probability of a fantastic meterological phenomenon

= P(all 6 rods are struck)

= 0.2⁶

= 0.000064 Answer 5

DONE