Question

1. We are exploring the following model for fireflies flashing: a firefly flips a biological equivalent of a coin every second. If it comes up heads (probability of 0.1) it lights up for that second; otherwise it does not.
   1. For a given firefly, what is the average number of flashes over a minute? What is the spread (standard deviation) of that number?
   2. If we have 1000 fireflies in our field of view, what is the expected number of flashes we will see in 10 seconds?
   3. If each flash is .1 lumens and we want 40 lumens at least 70% of the time, how many fireflies do we need?
   4. We are monitoring 100 fireflies and determine that the number of flashes per minute per firefly is about 7. Is this consistent with our model?

Answer 1

This is a binomial distribution as it has only two outcomes.

Probability of success = p = 0.1

a) n = 60 (as a minute has 60 seconds, so 60 trials)

Say X be the number of flashes

X~ B(60, 0.1)

E(X) = np = 60*0.1 = 6

V(X) = √(npq) = √(60*0.1*0.9) = 2.32

b) Total number of trials = 1000*10 = 10000

X~B(10000, 0.1)

E(X) = 10000*0.1 = 1000

c) If each flash is 0.1 lumens, 40*10 flashes OR 400 flashes equal 40 lumens

P(X>400) = 0.7 (given information)

P(X < 400) = 0.3

P(Z < z0) = 0.3 when z0 = -1.88 (from the z-table)

z0 = -1.88 = (400 - E(X))/V(X)

-1.88V = 400 - E

-1.88 √(n*0.1*0.9) = 400 - n*0.1

-1.88*0.3√n = 400 - 0.1n

0.1n - 0.564√n - 400 = 0

n should be 4373

d) According to our model, E(X) = n*p = 60*0.1 = 6

Hence, it is not consistent with our model

n =