Question

2. We derived in class the expected time until discovery of a single firm (R&D - expected discovery) Using the same set-up, let two firms engage in R&D. (a) What is the probability that neither of the firms discovers at a particular date? b) What is the probability that at least one of the two firms discovers at a particular date? (c) Calculate the expected date of discovery (ie, Ea (T)).
The second a,b,c is the solutions. I was wondering the steps tp solve this question.

Answer 1

Solution:

The question misses to provide the basic set-up, for what's done in the class. Denoting alpha by p, for ease of writing. It seems like for any of the firm, probability of discovering something at a particular date is p.

Then,

(a) With probability p, a firm is able to discover at a particular time. Then, with probability (1- p), a firm do not discovers at a particular time. Further, note that discovery by one firm is independent of discovery by another firm. So,

Prob(neither firm discovers at a particular date) = Prob(firm 1 doesn't discover AND firm 2 doesn't discover)

= Prob(firm 1 doesn't discover)*Prob(firm 2 doesn't discover) (due to independence)

= (1 - p)*(1 - p) = (1 - p)²

(b) Finding probability that at least one firm discovers at a particular date:

Note that at least one firm simply implies either only firm 1, or only firm 2, or both the firms discover at a particular time.

Also, a firm has only two possible cases: discover and do not discover

So, jointly, only four possibilities occur: neither firm discovers, firm 1 discovers but not firm 2, firm 2 discovers but not firm 1, and both firms discover

So, total probability adds up to 1(as we know), we have, Pr(neither firm discovers) + Pr(only firm 1 does) + Pr(only firm 2 does) + Pr(both discover) = 1 ... (*)

We find Pr(at least one firm discovers) = Pr(only firm 1 discovers) + Pr(only firm 2 discovers) + Pr(both firms discover)

From (*), this becomes: Pr(at least one firm discovers) = 1 - Pr(neither firm discovers)

Pr(at least one firm discovers) = 1 - (1 - p)² (using part (a))

(c) To find the expected date of discovery, more information would be helpful.

Expected discovery date = sum(probability of discovery in time t*time t)

Also, does E₂(T) denotes expected discovery date for only firm 2.