Brain Teasers are Back!

I agree, Eoink’s description is rather elegant.

SteveD’s answer was spot-on and no doubt based on some rather elegant working as well.

Cheers
Don

I’ve been keeping weather records for Newbury for some years (sad, but not strictly true). I have records which show that for the month of April, if it’s fine today, the chances are 3/4 that it will be fine tomorrow. If it’s wet today, the chances are 2/3 it will be wet tomorrow.

Today is Tuesday and it’s fine and sunny. I intend to go for a walk on Friday. What is the probability that Friday will be a fine day?

For those of us who like “elegant” solutions, the above teaser is not for you !!

The solution is straightforward enough and the idividual elements (at least, the way I did it) are satisfying in themselves, But … the final summing up is anything other than elegant ! In fact, IMHO, it’s disappointingly inelegant, ie ugly !

I’ll try a response, 52.3%
Tuesday: Fine
Wednesday: 3/4 chance fine, 1/4 chance wet
Thursday: Fine (3/4 x 3/4) + (1/4 x 1/3) = 6/16 + 1/12 = 0.458. Wet = 1 - 0.485 = 0.542
Friday: Fine: (0.458 x 0.75) + (0.542 x 1/3) = 0.343 + 0.181 = 0.523, or 52.3%

That’ a mighty fine shot at it Mike, but …

… you are correct that it’s three days, Wed, Thurs, Fri
and, that you need to follow the probabilities each day up to and including Friday.

But even if it’s Wet on Wednesday (or Thursday) there is a possibility of it being fine the following day.

When I did the arithmetic using vulgar fractions, the probabilities on each branch of the “tree” diagram looked quite neat (there were four of them). But … when I came to add them up, the lowest common denominator that I could find was 576

Hmm, I thought I did that…

Worth drawing a probability tree, there are a few examples in this thread.

With just two choices (wet or fine) and three days, there are eight possible “roots” to Friday. Four roots will lead to “fine on Friday” and four will lead to “wet on Friday”

All eight roots will add up to a probability of “one” (assuming the sums are right). The four leading to “fine in Friday” will add up to … x/576

Here is a picture of a probability tree.

Hopefully a picture is worth a thousand words ?

Yes, I didn’t do the tree, just in my head. But I have 4 roots for fine and 4 for wet. I took a shortcut with subtracting fine from 1 to get the wets, but there’s 8 probabilities in there. It’s entirely possible I made an error of course.

Too late for probability trees here, it’s beer and music :sunglasses:

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Wise move, just realised it must be midnight or thereabouts.

Here in the UK, we are in the middle of another sunny day - weatherwise ! Almost time for lunch and a beer as well !

Cheers
Don

I’ve just realised, if it’s in England in April it will snow at Easter and be wet every day…

Ha, ha, nice try Mike !!

Well Mike, i’m going to give you nine and three quarters out of ten.
Because you did it in your head, I should give you an extra mark, but that would mean you get more than 10 out of 10 so i’m going to be really churlish and stick with nine and three quarters !!

Why ?

Well, you certainly got the principles right and most of the figures, but one little mistake in the arithmetic, (you’ll kick yourself), got in the way :sunglasses:

I’ve highlighted it for you. In bold italics. 3/4 x 3/4 = 9/16 not 6/16 … (groan !)

When you work out the arithmetic you should then get 31/48 for it being Fine on Thursday and hence 17/48 for it being Wet on Thursday.

Friday then brings you in for fine weather as (31/48 x 3/4) + ( 17/48 x 1/3)

Which leads you to 347/576 or 60% more or less !

Well done !

And because you did it in your head, let’s go 10/10 :sunglasses:

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You know Mike, you really did do well to sort out the weather for Friday. I didn’t really expect anybody to sort through that tangled web of numbers !!

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My Probability Tree, which is a slightly different approach to Mike’s, but no less fallible !

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Groan, indeed. It was late so I didn’t go back and check the arithmetic. It’s one of those errors, that when you make it, you keep on reading it as being right!

I know the feeling Mike.

It took me about three readings of your post before I noticed what was wrong. Your version just LOOKED so right !

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I bet that Eoink, Dozey and one or two others were impressed with your solution, despite the trivial goof ?

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I should warn you that I got 100% in school certificate maths. But it was a long time ago now…

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I was very impressed by that, very quick and neat.

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