A courier service company sends 30% of its orders by air, 50% by combination of bus and local transport and remaining 20% by train. Past record shows the courier is delivered late 2%, 7% and 5% of the time when orders are sent by air, bus local transport and train respectively. Find (i) the probability that the order will be delivered late (ii) the probability that the parcel delivered to a customer is sent by train if it is delivered late.
Answers
Step-by-step explanation:
The restrictions put in place to limit the diffusion and impacts of Covid-19 have had a widespread impact on people’s lives, and the way energy is used across entire economies.
One of the biggest impacts has been the reduction in passenger transport demand, due to a combination of government lockdowns and fears of contracting and spreading the virus when using mass transport modes. While freight transport has also been reduced, the drivers of freight activity during the current crisis are complex, driven by both supply- and demand-side factors, and in the latter, by the need to keep essential services operating. In contrast, passenger transport, (for both leisure and business travel) is often optional, and more influenced by people’s decision-making processes. The focus of this paper is therefore on passenger transport.
The probability that the order will be delivered late = 0.051
The probability that the parcel delivered to a customer is sent by train if it is delivered late = 0.196
Given:
i) A courier service company sends 30% of its orders by air, 50% by combination of bus and local transport and remaining 20% by train.
ii) The courier is delivered late 2%, 7% and 5% of the time when orders are sent by air, bus local transport and train respectively.
To find:
(i) the probability that the order will be delivered late
(ii) the probability that the parcel delivered to a customer is sent by train if it is delivered late.
Solution:
Let the events be defined as:
A: Courier service company sends its orders by air
B: Courier service company sends its orders by bus local transport
T: Courier service company sends its orders by train
L: The courier is delivered late
It is given that
P(A) = 0.3
P(B) = 0.5
P(T) = 0.2
P(L/A) = 0.02
P(L/B) = 0.07
P(L/T) = 0.05
i) The probability that the order will be delivered late = P(L)
By Total Probability theorem, we can say
P(L) = P(L n A) + P(L n B) + P(L n T)
=> P(L) = P(A).P(L/A) + P(B).P(L/B) + P(T).P(L/T)
=> P(L) = 0.3*0.02 + 0.5*0.07 + 0.2*0.05
=> P(L) = 0.3*0.02 + 0.5*0.07 + 0.2*0.05
=> P(L) = 0.006 + 0.035 + 0.01
=> P(L) = 0.051
ii) the probability that the parcel delivered to a customer is sent by train if it is delivered late = P(T/L)
By Bayes' theorem, we can say
P(T/L) = P(T).P(L/T)/P(L)
=> P(T/L) = 0.2*0.05/0.051
=> P(T/L) = 0.01/0.051
=> P(T/L) = 10/51 = 0.196
Hence,
the probability that the order will be delivered late = 0.51
the probability that the parcel delivered to a customer is sent by train if it is delivered late = 10/51 = 0.196
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