Question

You're tossing 1000 paper airplanes off the roof onto a field, trying to hit a 1m square target. The airplanes are independent. The probability of any particular airplane hitting the target is 0.1%. The random variable X is the number of airplanes hitting the target.

1. What's the best probability distribution for X?

2. (5) Name another distribution that would work if you computed with very large numbers.

3. (5) Name another distribution that does not work in this case, but would work if the probability of any particular airplane hitting the target is 10%

4. You want to test a suspect die by tossing it 100 times. The number of times that each face from 1 to 6 shows is this: 12, 20, 15, 18, 15, 20.

   a. (5) What's the appropriate distribution?
   b. (5) If the die is fair, what's the probability that the observed distribution could be that far from the actual probability?

Answer 1

1) a) X~Binomial(1000,0.001) distribution

Pdf of X: P(X=x) = 1000Cx*0.001^x*0.999^1000-x

Where x=0,1,.....,1000

b) Another distribution with p=0.10, and n=1000

np=1000*0.10= 100, √np(1-p) = 9.487

So, X~Normal(100,9.487) distribution here (from the Central limit theorem)

2) a) Probability of getting any number on a fair die= 1/6

We have n=100

X~Binomial(100,1/6)

b) From the observed numbers, the farthest is 20,

So Probability of this is: P(x=20) = 100C20*(1/6)^20*(5/6)^80 = 0.0679