A Letter Problem

Probability Level 3

A writes a letter to his friend B and gives it to his son to post it in a letterbox , the reliability of the son being $\frac{3}{4}$ . The probability that a letter posted will get delivered at all is $\frac{8}{9}$ . At a later date A hears from B that the letter had not reached him.

Find the probability that the son did not post the letter at all.

The answer is 0.75.

2 solutions

Brian Charlesworth
Jan 14, 2015

The probability that the letter does not reach $\textbf{B}$ is

P(son never posts letter) + P(son does post letter but the postal service fails) =

$\dfrac{1}{4} + \dfrac{3}{4}*(1 - \dfrac{8}{9}) = \dfrac{1}{3}$ .

So the conditional probability that the son did not post the letter at all given that the letter does not reach $\textbf{B}$ is

$\dfrac{\frac{1}{4}}{\frac{1}{3}} = \dfrac{3}{4} = \boxed{0.75}$ .

I must really salute you sir !!! . I had just posted one question and was working on another only to come back and see that there was a solution waiting for me .

I am a great fan of yours sir .

A Former Brilliant Member - 6 years, 5 months ago

Thanks, Azhaghu. You're posting good questions; keep up the good work. :)

Brian Charlesworth - 6 years, 5 months ago

We need to start a "Brian Charlesworth Fan Club" haha.

Trevor Arashiro - 6 years, 4 months ago

@Trevor Arashiro – Hahaha. No, not happening. :) Anyway, I appreciate the sentiment.

By the way, how's the new trig book? And those fancy new white boards?

Brian Charlesworth - 6 years, 4 months ago

@Brian Charlesworth – Loving the white board. I can now read my messy hand writing since it's in big font.

As for the trig book, I haven't used it much as I just had finals and it came in the mail about 2 weeks ago. Also I have to practice for the AMC 10 since I really want to make the AIME this year. I'm sure that I'll learn a lot from it however.

btw, just curious do you ski? I love skiing and the major down side of living in Hawaii is that there is no snow :'( #problemsinparadise

Trevor Arashiro - 6 years, 4 months ago

@Trevor Arashiro – I didn't know anything about the AMC or AIME so I did some reading. Looking at some sample exams it would seem that the questions are similar to the more difficult ones found on Brilliant, so I'd imagine that some of your practice time is spent on this site. Judging from your output here I would say that scoring in the top 2.5% on the AMC 10 is well within reach. Are you pretty good at avoiding 'silly' mistakes? At your age I had a tendency to make those, so I had to learn to focus on being deliberate when it came to the details. Good luck with making the AIME. :)

As for skiing, there are ski hills within a few hours drive, but I've never been much of a thrill-seeker. The climate on southern Vancouver Island is temperate, so the winter experience is quite different than it is for most of Canada. We only had one sprinkling of snow back in November and nothing since. So in general the winter sports scene is non-existent here.

But I googled "skiing in Hawaii" and found that there is some skiing possible on Mauna Kea! Who'da thunk? No lifts or anything, but some nice powder, (weather permitting), and an amazing view. That would be quite the experience. The next closest hill, though, would be Mt. Shasta, about 4000 km away, so yeah, I guess you do have a #probleminparadise. :)

Brian Charlesworth - 6 years, 4 months ago

@Brian Charlesworth – Thank you so much for the encourage menta. I've done 3 practice exams. 75 mins each over the past week. Got a 146, 130.5 (this is the one where I was rally strict on timing and everything, and 136. You need a 120 to get to the AIME, so if I can play like I practice I think I'm good. I don't think I can quite make the USAMO, that's a little out of my reach at the moment.

As for careless mistakes. In Trevor M Arashiro, my middle name is mistakes (it's actually Matthew, which can be rearranged to wet math :P ). On the one where I got a 130.5, I literally skipped a question because I didn't see it, that cost me 4.5 points. And I thought the radius of a circle was 4 when its diameter was actually 4, so that cost me 6. I left 2 blank, one of which was combinatorics, which isn't my strong suit, and the other was sets where I couldn't understand the question. In total, I should have had a 142. And on the one where I got a 132, I put $46*2=102$ cuz logic -.-

Trevor Arashiro - 6 years, 4 months ago

@Trevor Arashiro – Hahaha. The radius/diameter mistake is so easy to make; it's still the most common one for me. Your scores are great despite the minor slip-ups, so you don't need to give yourself too much grief about making 1 or 2 of them. I'm sure that by 'game day' all your practice will pay off. :)

Brian Charlesworth - 6 years, 4 months ago

Mudit Bansal
Jan 18, 2015

$p(required)=\quad \frac { p(son\quad did\quad not\quad post\quad the\quad letter) }{ p(son\quad did\quad not\quad post\quad the\quad letter)+p(son\quad posted\quad it\quad but\quad not\quad delivered) } \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\quad \frac { \frac { 1 }{ 4 } }{ \frac { 1 }{ 4 } +\frac { 3 }{ 4 } *\frac { 1 }{ 9 } } \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad =\quad \frac { 3 }{ 4 }$

A Letter Problem

The answer is 0.75.

2 solutions

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