Question

A commuter must pass through five traffic lights on her way to work and will have to stop at each one that is red. She estimates the probability model for the number of red lights she hits (x), as shown below: x 0 1 2 3 4 5 p(x) 0.04 0.23 p 0.1 0.1 0.1 a)Find the probability that she hits at most 3 red lights. Answer to 2 decimal places. b)Find the probability that she hits at least 3 red lights. Answer to 2 decimal places. c)How many red lights she expect to hit? Answer to 2 decimal places. d)What is the standard deviation of number of red lights she hits? Answer to 3 decimal places. e)Let us consider any two consecutive days. What is the chance that she hits exactly two red lights on both days? Answer to 4 decimal places

Answer 1

Ans:

As,all probabilities sum upto 1 for a valid probability distribution.

P(x=2)=1-(0.04+0.23+0.1+0.1+0.1)=1-0.57=0.43

x	p(x)
0	0.04
1	0.23
2	0.43
3	0.1
4	0.1
5	0.1
Total=	1

a)P(atmost 3)=P(x<=3)=0.04+0.23+0.43+0.1=0.80

b)P(atleast 3)=P(x>=3)=0.1+0.1+0.1=0.30

c)Expected hits=0*0.04+1*0.23+2*0.43+3*0.1+4*0.1+5*0.1=2.29

d)standard deviation

x	p(x)	x*p(x)	(x-2.29)^2*p(x)
0	0.04	0	0.2098
1	0.23	0.23	0.3827
2	0.43	0.86	0.0362
3	0.1	0.3	0.0504
4	0.1	0.4	0.2924
5	0.1	0.5	0.7344
Total=	1	2.29	1.7059
		std. dev.=	1.306

Variance=1.7059

standard deviation=sqrt(1.7059)=1.306

e)

P(x=2)*P(x=2)=0.43*0.43=0.1849