Parcels from sender S to receiver R pass sequentially through two post-offices. Each post-office has a probability \(\frac{1}{5}\) of losing an incoming parcel, independently of all other parcels. Given that a parcel is lost, the probability that it was lost by the second post-office is_________.
Concept:
\(p\left( {\bar A} \right) = 1 - p\left( A \right)\)
Conditional Probability: The probability of an event A given that event B has occurred is defined as:
\(p\left( {\frac{A}{B}} \right) = \frac{{p\left( {A\; \cap B} \right)}}{{p\left( B \right)}}\)
Calculation:
Probability of losing a parcel = 1/5
Probability of successfully passing a parcel = 1 - Probability of losing a parcel
= 1-1/5 = 4/5
There are two cases for losing the parcel from sender S to receiver R.
Lost at first post-office:
Probability of losing the parcel at first post-office \( = \frac{1}{5}\)
Successfully passed at first post-office but lost at 2nd post-office:
Probability of losing the parcel corresponding to this case will be:
\( = \frac{4}{5} \times \frac{1}{5}\)
Net probability of losing the parcel will be:
\( = \frac{1}{5} + \frac{4}{5} \times \frac{1}{5} = \frac{9}{{25}}\)
\(p\left( {\frac{{Parcel\;lost\;at\;2nd\;post\;office}}{{parcel\;lost}}} \right)\)
\( = \frac{{p\left( {Parcel\;lost\;at\;2nd\;post\;office\; \cap \;parcel\;lost} \right)}}{{p\left( {parcel\;lost} \right)}}\)
\( = \;\frac{{p\left( {parcel\;lost\;at\;2nd\;post - office} \right)}}{{p\left( {parcel\;lost} \right)}}\)
\( = \frac{{4/25}}{{9/25}} = \frac{4}{9} \approx 0.44\)
Important Questions from Conditional Probability
Two events A and B are such that P(not B) = 0.8, P(A ∪ B) = 0.5 and P(A|B) = 0.4. Then P(A) is equal to
For two events, A and B, it is given that \({\rm{P}}\left( {\rm{A}} \right) = \frac{3}{5},{\rm{\;P}}\left( {\rm{B}} \right) = \frac{3}{{10}}\) and \({\rm{P}}\left( {{\rm{A|B}}} \right) = \frac{2}{3}\) . If A̅ and B̅ are the complementary events of A and B, then what is P(A̅ | B̅) equal to?
For two mutually exclusive events A and B, P(A) = 0.2 and P (A̅ ∩ B) = 0.3. What is P (A|(A ∪ B)) equal to?
If an event B has occurred and has P(B) = 1, the conditional probability P(A|B) is equal to:
If P(A) = 0.7, P(B) = 0.5 and P(B/A) = 0.3, find (i) P(A/B) (ii) P(A ∪ B)?
