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A fair die bearing the numbers 1, 2, 3, 4, 5, 6 is repeatedly thrown
until the running total first exceeds 12. What’s the most likely total
that will be obtained?
This must be the most likely total!
Here is the question!
Just consider the the next-to-last throw where last throw would make total exceeding 12.
Till then total must be either of 12,11,10,9,8 or 7. And next throw will make total greater than 12.
At this point -
If total is 12 then the first total exceeding 12 could be any from 13,14,15,16,17,18.
If total is 11 then the first total exceeding 12 could be any from 12,13,14,15,16,17,
If total is 10 then the first total exceeding 12 could be any from 12,13,14,15,16.
If total is 9 then the first total exceeding 12 could be any from 12,13,14,15.
If total is 8 then the first total exceeding 12 could be any from 12,13,14.
If total is 7 then the first total exceeding 12 could be any from 12,13.
It's quite evident that the number 13 appears in every above case.
Hence, 13 is the most likely total.
A man stuck in a small sailboat on a perfectly calm lake throws a stone overboard. It sinks to the bottom of the lake.
When the water again settles to a perfect calm, is the water level in the lake higher, lower, or in the same place compared to where it was before the stone was cast in?
Did you think it will rise?
But why water level was affected?
Do you recall what does Archimedes Principle state? For an object to float on water, it has to displace that much volume of water whose weight is equal to weight of the object itself. Now if object has less density than water then obviously it has to displace lesser amount of volume of water to float on it. That means it has to sink less in water.
For a moment, let's assume the stone has very high density & hence having weight equal to hundreds of kilograms despite of having small volume.
Here, stone sinks to the bottom of the lake suggests that it is has more density than water. It can't displace the water whose weight is equal to it's weight.But when it was on sailboat it could push the sailboat down so that more water is displaced weighing equal to it's weight. Result of this, the sailboat sinks little 'deeper' compared to when stone wasn't there.
Obviously, the volume of displaced water when stone was in sailboat (due to stone only) must be greater that the volume of displaced water when stone sinks to the bottom of the lake. That's why both sailboat and stone together could float on the water. In short, sailboat helped stone to displace amount of water needed to float which results in rise in shoreline.
And when stone is thrown out of the sailboat, then ideally it can't displace more water than when it was on sailboat. Now, sailboat sinks less 'deeper' in water displacing only water need to float itself.
That's why the water level must be dropped compared to earlier. The little rise due to water displaced by stone can't exceed the earlier water level for the reason explained above.
