Question

please explain, why and which formula are used and explanation, thanks

1. what is the probability that five guests of a twenty-floor hotel all stay on different floors (no pair of them are on the same floor) if, a. Any guests can stay on any floor? b. Two smokers of the group must stay in four smoking floors and the rest must stay in sixteen non-smoking floors.

2. Two dice ( one black and one white) are rolled simultaneously five times. What is the probability that number on the black dice be larger than on the white dice more than twice?

3. Suppose X and Y are two real numbers picked randomly and independently between 0 and 1. find [ XY greater than equal to 0.25 ].

Answer 1

1.

There are total 24 floor so there are total 24*23*...*20 = 5100480 ways.

Using 2nd condition two smokers of the group can stay in P(4,2) ways and remaining in P(16,3) ways.We use P(permutation) since no repeatation is allowed.

So desired probability is:P(4,2)*P(16,3)/5100480=12*3360/5100480=0.007905138

2.

The sample space S of two dice is shown below.
S = { (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(6,1),(6,2),(6,3),(6,4),(6,5),(6,6) }

E:number on the black dice be larger than on the white dice

n(E) = 16,n(S)= 36

P(E) = n(E)/n(36) = 16/36 = 0.44

3.

0<x<1,0<y<1

P[XY>=0.25] = P[X>0.25/Y] = ?

f(x,y) = f(x)*f(y) since x and y are ind.

= 1/(1-o)*(1/(1-0) = 1

P[X>0.25/Y]

cy= 10.25yfy = 0.25/2 = 0.125 J J0.25y