Two-ships-have-the-same-berth-in-a-port-It-is-known-that-the-arrival-times-of-the-two-ships-are-independent-and-have-the-same-probability-of-docking-on-a-Sunday-00-00-24-00-If-the-berth-

Question Number 207752 by efronzo1 last updated on 25/May/24

Two ships have the same berth

in a port . It is known that the

arrival times of the two ships

are independent and have the

same probability of docking

on a Sunday (00.00 - 24.00)

If the berth time of the first ship

is 2 hours and the berth time

of the second ship is 4 hours,

the probability that one ship

will have to wait until the

berth can be used is

\frac{67}{144} \frac{67}{288} \frac{1}{4} \frac{33}{144}

Commented by mr W last updated on 25/May/24

p = 1 - \frac{22^{2} + 20^{2}}{2 \times 24^{2}} = \frac{67}{288} ✓

Commented by efronzo1 last updated on 25/May/24

why sir 1 - \frac{22^{2} + 24^{2}}{2 . 24^{2}} ?

Commented by mr W last updated on 25/May/24

m y m e t h o d s e e b e l o w

Answered by mr W last updated on 25/May/24

Commented by mr W last updated on 25/May/24

s a y t h e f i r s t s h i p a r r i v e s a t t i m e x

(0 ⩽ x ⩽ 24) a n d t h e s e c o n d s h i p a t

t i m e y (0 ⩽ y ⩽ 24) .

s u c h t h a t t h e s e c o n d s h i p {d o e s n}^{'} t

n e e d t o w a i t :

y ⩾ x + 2 \dots (i)

s u c h t h a t t h e f i r s t s h i p {d o e s n}^{'} t

n e e d t o w a i t :

x ⩾ y + 4 \dots (i i)

w e c a n i n t e p r e t t h i s g e o m e t r i c a l l y :

e a c h p o i n t (x, y) i n t h e s q u a r e

24 \times 24 s h o w s t h e r a n d o m a r r i v a l

t i m e s o f b o t h s h i p s . i f t h i s p o i n t l i e s

i n t h e h a t c h e d z o n e s, w h i c h m e a n

t h e c o n d i t i o n s (i) a n d (i i), t h e n

n o s h i p n e e d s t o w a i t . o t h e r w i s e

o n e s h i p h a s t o w a i t .

t h e a r e a o f h a t c h e d z o n e s i s

\frac{22^{2} + 20^{2}}{2} . t h e t o t a l a r e a i s 24^{2} .

s o t h e p r o b a b i l i t y t h a t o n e s h i p

h a s t o w a i t i s

p = \frac{24^{2} - \frac{22^{2} + 20^{2}}{2}}{24^{2}} = 1 - \frac{22^{2} + 20^{2}}{2 \times 24^{2}} = \frac{67}{288}

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