Suppose that a van with capacity for 7 passengers departs from a commuter station

Question: Suppose that a van with capacity for 7 passengers departs from a commuter station. Observation has shown that the bus never departs empty. Let X denote the number of passengers that are female and Y the number of passengers that are male. Assume that all possible (x, y) pairs are equally likely. (a) Find the probability that the number of female passengers is more than 5. (a) Find the probability that the total number of passengers is less than 5.

Solution:

pairs of passengers:-
{(0,7),(0,6),(0,5),(0,4),(0,3),(0,2),(0,1)
(1,6),(1,5),(1,4),(1,3),(1,2),(1,1),(1,0)
(2,5),(2,4),(2,3),(2,2),(2,1),(2,0)
(3,4),(3,3),(3,2),(3,1),(3,0)
(4,3),(4,2),(4,1),(4,0),
(5,2),(5,1),(5,0)
(6,1),(6,0)
(7,0)}
p(x,y) = 1/35 where x = no of female and y = no of males

a. Probability of female passengers is more than 5
possible outcomes = (6,0),(6,1),(7,0)
P(A) = 3/35 = 0.0857

b. Passenger is less than 5
possible outcomes = {(4,0),(3,0),(3,1),(2,2),(2,1),(2,0),
(1,3),(1,2),(1,1),(1,0),(0,3),(0,2),(0,1),(0,4)}
= 14
P(B) = 14/35 = 0.4

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