Context: Conditional Probability

Question: You travel from New York to Singapore, with connecting flights in Chicago and Japan. You have one checked in luggage that is transferred from one airplane to another. In New York, there is a 5% chance that your luggage is not placed in the right airplane. The probability is 3% in Chicago and 2% in Japan. What is the probability that your luggage does not reach Singapore with you? Given it does not reach Singapore, what is the probability that it was lost in Chicago?

1) What is the probability that your luggage does not reach Singapore with you?

To get the probability that your luggage does not reach Singapore with you, we merely add up the total probability of your luggage being lost in New York, in Chicago, and in Japan. Your flight path to your destination is from New York ?→? Chicago ?→? Japan ?→? Singapore. The probability of losing your luggage in New York is simply 0.05. The probability of losing your luggage in Chicago is equal to the probability of not losing it in New York (0.95), times the probability of losing it in Chicago (0.03). The probability of losing your luggage in Japan is equal to the probability of not losing it in New York (0.95), times the probability of not losing it in Chicago (0.97), times the probability of losing it in Japan (0.02).

Solution:

?0.05+(0.95)(0.03)+(0.95)(0.97)(0.02)=? 0.09693

2) Given it does not reach Singapore, what is the probability that it was lost in Chicago?

We can solve for the probability of losing your luggage in Chicago given it does not reach Singapore using the Bayes' Theorem.

The Bayes' Theorem is given by: ?P(A?B)=P(B)P(B?A)×P(A)??. To solve for the probability of A given B, we have to get the probability of B given A times the probability of A, divided by the probability of B.

?P(A?B)? = The probability that the luggage was lost in Chicago, given it did not reach Singapore

??P(B?A)? = The probability of luggage not reaching Singapore, given it was lost in Chicago. (equals to 1 since the luggage 100% does not reach Singapore if it was lost in Chicago)

?P(A)=? Probability of losing luggage in Chicago

?P(B)=? Probability of your luggage not reaching Singapore (equals to answer in 1))

Solution:

??P(A?B)=0.09693(1)(0.95×0.03)?=? 0.294

The probability of losing your luggage in Chicago given that it does not reach Singapore is 0.294