The gift problem

Question

I have a probability problem that I do not have any insight on how to solve it. Here is the problem:

There are three packages in the post office for you. It is Christmas and you know that there is a probability of 100% to a present be sent to you from your mother, there is a probability of 40% to a present be sent to you from your girlfriend and there is a probability of 1% to a present be sent to you from others.

Given that at the post office you hold one of the three packages that you know that it was not been sent by your girlfriend, answer:

1) What is the probability of the gift has been sent by your mother? 2) What is the probability that the others two presents have been sent by: 2.1) one from your mother and other from you girlfriend? 2.2) two from your girl friend? 2.3) two from an other person?

I would appreciate any tip or insight about the solution of this problem.

Thanks

Edited:

Here is what I think about the line thinking to solve the problem. First, we could start from a simpler scenario, we could image that there is just one package in the post office. In this situation, given that there is just one package, for sure the present was been sent by your mother and not by your girlfriend or other sender. Great! So, the next scenario could be the one with two packages. In it, for sure, one present would be sent by the mother and the other present would be sent by the girlfriend or by a other sender or by the mother again with the specified probabilities. Now, we could think in the scenario with three packages.

Answer 1

Restating the problem:

You are notified upon the arrival of three packages at the post office intended for you. It is Christmas eve and you expect a gift from your mother and another from your girlfriend. The third package, however, you don't have a clue about who's the sender. Suppose that the chance of you receiving a gift from your mother is 100%, from your girlfriend 40% and the chance of you getting a package from a third party is 1%.

After arriving at the post office you grab the first package and, although you can't tell who the sender is, you know it wasn't your girlfriend.

1. What's the probability that the package you grabbed was sent by your mom?

2. What's the probability that the other 2 remaining packages were sent by:

   • 1 by your mom, 1 your girlfriend;
   • 2 by your girlfriend;
   • 2 by a third party;

Answer 2

Is this problem your invention?

As you wrote "Now, we could think in the scenario with three packages", it seems an invention, and as stated in the comments, not at all clear.

So, let's go on with the following tips or insights.

First scenario

There is just one package sent by your mother. EASY.

Second scenario

There are two packages: one sent by your mother and the other sent by your girlfriend or other sender.

For the second package we have to rescale the values in order to get 1.

(40)/(1) -> (40/41)/(1/41)

Here we have two options:

• MOTHER-GIRLFRIEND
• MOTHER-OTHER SENDER

Third scenario

There are three packages: one sent by your mother, another sent by your girlfriend or other sender, and the third sent by you mother, your girlfriend or other sender.

For the second package we have to rescale the values in order to get 1.

(40)/(1) -> (40/41)/(1/41)

For the third package we have to rescale the values in order to get 1 also.

(100)/(40)/(1) -> (100/141)(40/141)/(1/141)

Here we have six options:

• MOTHER-GIRLFRIEND-MOTHER
• MOTHER-GIRLFRIEND-GIRLFRIEND

• MOTHER-GIRLFRIEND-OTHER SENDER

• MOTHER-OTHER SENDER-MOTHER

• MOTHER-OTHER SENDER-GIRLFRIEND
• MOTHER-OTHER SENDER-OTHER SENDER

Fourth scenario

There are three packages: each of them can be sent by your mother, your girlfriend or other sender.

For each package we have the following probabilities (again rescaling):

(100)/(40)/(1) -> (100/141)(40/141)/(1/141)

Here we have ... options:

• MOTHER-MOTHER-MOTHER
• MOTHER-MOTHER-GIRLFRIEND
• MOTHER-MOTHER-OTHER SENDER
• ....

See if this helps you.