Step by step procedure.

Question

consider an experiment in which each event of four cars exiting from a university entrance turns right (R) or left (L) . Let the random variable X represent  the number of cars which turned left. Assume that a car will turn right or left with equal probability of 1/2.

A. What is the sample space S?

B. What are the possible values that X can take?

C. Develop the probability distribution for X.

D. Find the mean and standard deviation of the distribution of X.

E. Compute P(x>0)

Answer 1

The total of different results is 16:

RRRR, RRRL, RRLR, RRLL, RLRR, RLRL, RLLR, RLLL, LRRR, LRRL, LRLR, LRLL, LLRR, LLRL, LLLR, LLLL.

The number of cars turning left, X, has 16 corresponding values:

0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4.

We can write these as frequencies (in brackets): 0 (1), 1 (4), 2 (6), 3 (4), 4 (1). This is a binomial distribution and the sequence of frequencies is 1 4 6 4 1, the coefficients of the binomial expansion of (p+(1-p))^4, and the fourth row of Pascal's Triangle (p=0.5 in this case).

The mean is (1*0+4*1+6*2+4*3+1*4)/16=32/16=2=np=4(1/2), where n is the number of cars.

The standard deviation is sqrt(((-2)^2+4(-1)^2+6*0+4(1)^2+(2)^2)/16)=sqrt((4+4+0+4+4)/16)=4/4=1=np(1-p)=4(1/2)(1/2).

P(x>0)=15/16 because there is only 1 zero in all possible 16 scenarios and there is a 50:50 chance of turning left or right.