Question

A seed randomly blows around a complex habitat. It may land on any of three different soil types: a high-quality soil that gives a 0.8 chance of seed survival, a medium-quality soil that gives a 0.3 chance of survival, and a low-quality soil that gives only a 0.1 chance of survival. These three soil types (high, medium, and low) are present in the habitat in proportions of 30:20:50, respectively. The probability that a seed lands on a particular soil type is proportional to the frequency of that type in the habitat.

a. Draw a probability tree to determine the probabilities of survival under all possible circumstances.b. What is the probability of survival of the seed, assuming that it lands?

Answer 1

a) Attached.

b) Probabiltiy of survival = 0.35

Explanation:

a) The probability tree is attached as a picture.

This is constructed with the two events (landing and survival), with their associated probabilities.

The events are dependant, as the probability of survival depends on the outcome of the first event (the soil where the seed landed).

b) The probability of survival can be calculated as the sum of the probabilities that include the event "survives":

`P(s)=P(H\&s)+P(M\&s)+P(L\&s)\\\\P(s)=P(H)P(s|H)+P(M)P(s|M)+P(L)P(s|L)\\\\P(s)=0.3*0.8+0.2*0.3+0.5*0.1\\\\P(s)=0.24+0.06+0.05\\\\P(s)=0.35`

The first line adds the three joint events that end with survival (land on high quality soil H and survive S, for example).

The second line splits the probability but the survival is conditioned to the soil where the seed has landed. That is why it is a conditional probability.