F.3 Probability

1)
let t in minutes

t=0 t=7 t=20 t=27 t=40
||----------||-----------------------||----------||-----------------------||
F left C left F left C left F left

the bus he will take if he arrive at this interval
(refer to the time line)
C F C F
||----------||-----------------------||----------||-----------------------||


so required probability=7/20


2)
let the number be k it can be a prime number OR a even number
P(prime)=P(even)
(2+a)/6=(3+b)/6
where
a=1 if k is prime, =0 if k is not
b=1 if k is even, =0if k is not
since 2+a=3+b, a=1,b=0
so k is a prime number that is not even
so k is a prime number except 2


3)
let the number be k
note that, the special dice has faces 1,1,1,3,3,k
P(special dice showing an '1')=0.5
let X be the number obtained after throwing the fist (normal) dice
If an '1' is obtained from the second (special) dice,
1+X>1*X

Hence
P(sum>product) is at least 0.5
since P(sum>product given that special dice showing an '1')=0.5
(as P(special dice showing an '1')=0.5)

Also notice that the first dice has an "1" and the second dice has "3,3,k"
1+3>1*3 and 1+k>1*k

adding these favourable out comes,
P(sum>product) must always >0.5
so you can put any number


see my summary
for the green box(that is the number k)
if k=1, P(sum>product)=26/36
if k>1, P(sum>product)=21/36


圖片參考：http://i256.photobucket.com/albums/hh182/zilu_photo/sshot-2010-01-01-23-54-19.png




圖片參考：http://i256.photobucket.com/albums/hh182/zilu_photo/sshot-2010-01-01-23-54-26.png