Question

1. A) If you flip an unfair coin 100 times, and the probability for a coin to be heads is 0.4, then the number of heads you expect on average is:

B) If you flip an unfair coin 100 times, and the probability for a coin to be heads is 0.4, then the standard deviation for the number of heads is:

C) If you flip an unfair coin 2 times, and the probability for a coin to be heads is 0.4, then what is the probability to find exactly 1 heads?

D) If you flip an unfair coin 2 times, and the probability for a coin to be heads is 0.4, then what is the probability to find exactly 2 heads?

E) If you flip an unfair coin 10 times, and the probability for a coin to be heads is 0.4, then what is the probability to find exactly 2 heads?

F) Suppose that 20 molecules jump randomly in and out of a cell. Eventually, each molecule is independently distributed and has a probability 0.1 to be in the cell.

What is the variance of the count of these molecules inside of the cell?

Answer 1

A) If you flip an unfair coin 100 times, and the probability for a coin to be heads is 0.4, then the number of heads you expect on average is:

Average heads = n*p = 100*0.4 = 40

B) If you flip an unfair coin 100 times, and the probability for a coin to be heads is 0.4, then the standard deviation for the number of heads is:

Standard deviation = (n*p*q)^0.5 = (100*0.4*0.6)^0.5 = 4.9

C) If you flip an unfair coin 2 times, and the probability for a coin to be heads is 0.4, then what is the probability to find exactly 1 heads?

P(X=1) = C(2,1)*0.4*0.6 = 2*0.4*0.6 = 0.48

D) If you flip an unfair coin 2 times, and the probability for a coin to be heads is 0.4, then what is the probability to find exactly 2 heads?

P(X=2) = 0.4*0.4 = 0.16

E) If you flip an unfair coin 10 times, and the probability for a coin to be heads is 0.4, then what is the probability to find exactly 2 heads?

P(X=2) = C(10,2)*(0.4^2)*(0.6^8) = 45*(0.4^2)*(0.6^8) = 0.1209

F) Suppose that 20 molecules jump randomly in and out of a cell. Eventually, each molecule is independently distributed and has a probability of 0.1 to be in the cell.

What is the variance of the count of these molecules inside of the cell?

Variance = n*p*q = 20*0.1*0.9 = 1.8