1 Q1) Both undergraduates and postgraduates can use the university cafeteria. Each diner can choose...

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1 Q1) Both undergraduates and postgraduates can use the university cafeteria. Each diner can choose between buying a meal or bringing a packed lunch. (Everyone has exactly one meal each, no more and no less). The cafeteria offers a daily choice between a hot meal or a cold meal. A survey of undergraduate diners finds that 40% of them bring their own food. Overall, only 25% of the diners bring their own food. Postgraduates make up one fifth of the diners in the cafeteria.

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1 Both undergraduates and postgraduates can use the university Q1) cafeteria. Each diner can choose between buying a meal or bringing a packed lunch. (Everyone has exactly one meal each, no more and no less). The cafeteria offers a daily choice between a hot meal or a cold meal. A survey of undergraduate diners finds that 40% of them bring their own food. Overall, only 25% of the diners bring their own food. Postgraduates make up one fifth of the diners in the cafeteria. a) What is the probability that a diner is an undergraduate and has bought a meal? [8] b) What is the probability that someone who has bought a meal is a postgraduate? [8] c) Give two different examples from the question of mutually exclusive events. [4] Q2) Two types of vehicle use the university central bus station: Uni-link buses and National Express coaches. a) On average, a National Express coach arrives every two hours. What is the probability that only one bus will arrive in a four hour period? [5] b) On the U1 Uni-link bus route, buses either go to the airport or to the docks. Other Uni-link routes do not go to the airport. At the bus station, half of the U1 buses which arrive are going to the airport. At midday, 10% of the Uni-link buses are full. Of these particular buses, 30% are airport buses on the U1 route. U1 buses make up 40% of the Uni-link arrivals at the bus station. What is the probability that a bus is full at midday, given that it is an Airport U1 bus? [6] c) On average, a certain number, k, of National Express coaches go to Bournemouth every day. It is found that the probability of n coaches going - k . to Bournemouth in a day is e What is n?Q3) If P(A) = 0.2 and P(AnB) = 0.4 a) find (5) P(BA) b) Can P(B) = 0? Explain your answer. (5) c) What is the smallest value that P(AB)can take? Explain your answer. (5) d) What would P(AB)have to be, for the complement...

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David · Accepted Answer

Solution 1: 
a) P (P) = 1/5 or 0.2, P (U) = 4/5 or 0.8 
P (U/M) = P (U and M)/ P (M) 
P (U and B) = 0.40 
P (U and M) = (1 - 0.60)* (1 -0.75) = 0.45 
P (B) = 0.25 
b) P (B) = 1 – P (M) = 0.75 
P (M and P) = 0.2*0.75 
P (M/P) = P (M and P)/ P (P) 
              = 0.15/0.2 
               = 0.75 
c) P (U and B) and P (U and M) both are not equal to zero.    
Solution 2: 
a) We have λ =2/4 = 0.5
Using Poisson distribution,
P [X = 1] = e-λ λ x/x!
               = e-0.5 (0.5) 1/1!
               = 0.9098
b) P(full at midday | airport U1 bus)
= P(Airport U1 bus | full at midday) . P(full at midday) / P(airport U1 bus)
P(airport U1 bus) = P(AirportNU1 bus)
P(Airport N U1 bus) = P(Airport | U1 bus) * P(U1 bus)
P(Airport n U1 bus) = 0.5 x 0.4
=0.2
therefore P(airport U1 bus | full at midday = 0.3, P(full at midday) = 0.1
0.3 x 0.1 / 0.2 = 0.15
c) We have λ =2/4 = 0.5
Using Poisson distribution
e-λ λ n/n!= e-k
e-0.5 (0.5)n/n! = e-k
e-0.5 (0.5)n/n! = e-k
n = 1 
Solution 3: 
We have P ( ̅      (    1 – 0.2 = 0.8 
a)   (   ̅   
  ( ̅   
  ( ̅ 
 
                               = 0.4/0.8 
                               = 0.5 
b) If P (B) = 0, then P ( ̅    which is not considered to be certainly true. P ( ̅    B) is 
0.

1 Q1) Both undergraduates and postgraduates can use the university cafeteria. Each diner can choose between buying a meal or bringing a packed lunch. (Everyone has exactly one meal each, no more and...

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