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705-805 (Hard)|   Geometry|   Probability|      
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Bunuel
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In a common carnival game, a player tosses a penny from a distance ont 05 Mar 2021, 04:45
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In a common carnival game, a player tosses a penny from a distance onto the square surface of a table, which is 1 inch^2 in area. If the penny, which is 3/4 inch in diameter, falls entirely inside a square, the player wins; otherwise, he loses. If the penny lands on the table, what is the probability of winning?

A. 1/256
B. 1/32
C. 1/16
D. 1/8
E. 1/4

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C
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In a common carnival game, a player tosses a penny from a distance ont 05 Mar 2021, 10:24
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Bunuel
In a common carnival game, a player tosses a penny from a distance onto the square surface of a table, which is 1 inch^2 in area. If the penny, which is 3/4 inch in diameter, falls entirely inside a square, the player wins; otherwise, he loses. If the penny lands on the table, what is the probability of winning?

A. 1/256
B. 1/32
C. 1/16
D. 1/8
E. 1/4

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The square measures 1 inch by 1 inch. The radius of the penny is 3/8 of an inch. For the penny to land entirely within the square, the penny's center must be at least 3/8 of an inch from the left edge of the square, and at least 3/8 of an inch from the right edge of the square. Similarly it must land 3/8 of an inch from the top edge and 3/8 of an inch from the bottom edge. That only leaves a 1/4 by 1/4 inch square in the middle of the table on which the center of the penny can land.

From the answer choices, I'm guessing the question means to tell us that the center of the penny lands in a random spot somewhere on the table. If that's true, then the 'winning' area is 1/16 square inches, and the total area is 1 square inch, and the probability of winning is thus (1/16)/1 = 1/16.

But I'm just guessing that's what the question means from the answer choices. It would be perfectly reasonable reading the question to assume the penny 'lands' on the table if any part of the penny hits the table, even if it's just the edge of the penny hitting the edge of the table. Then as long as the center of the penny 'lands' no further than 3/8 of an inch from an edge of the table, it hits the table. Then there's a much larger square the penny can 'land' on -- it would measure 1 + 3/8 + 3/8 inches on each side, or 7/4 inches on each side, for a total area of 49/16 square inches. Then the probability of 'winning' is (1/16)/(49/16) = 1/49.
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In a common carnival game, a player tosses a penny from a distance ont 17 Mar 2021, 00:31
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Beautifully done.

Although you lost me a bit on the end lol.

I think you mean the center of the penny has to just hit the edge.....otherwise if it goes too far, it won’t be balanced and it will fall off?


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Bunuel
In a common carnival game, a player tosses a penny from a distance onto the square surface of a table, which is 1 inch^2 in area. If the penny, which is 3/4 inch in diameter, falls entirely inside a square, the player wins; otherwise, he loses. If the penny lands on the table, what is the probability of winning?

A. 1/256
B. 1/32
C. 1/16
D. 1/8
E. 1/4

Are You Up For the Challenge: 700 Level Questions: 700 Level Questions

The square measures 1 inch by 1 inch. The radius of the penny is 3/8 of an inch. For the penny to land entirely within the square, the penny's center must be at least 3/8 of an inch from the left edge of the square, and at least 3/8 of an inch from the right edge of the square. Similarly it must land 3/8 of an inch from the top edge and 3/8 of an inch from the bottom edge. That only leaves a 1/4 by 1/4 inch square in the middle of the table on which the center of the penny can land.

From the answer choices, I'm guessing the question means to tell us that the center of the penny lands in a random spot somewhere on the table. If that's true, then the 'winning' area is 1/16 square inches, and the total area is 1 square inch, and the probability of winning is thus (1/16)/1 = 1/16.

But I'm just guessing that's what the question means from the answer choices. It would be perfectly reasonable reading the question to assume the penny 'lands' on the table if any part of the penny hits the table, even if it's just the edge of the penny hitting the edge of the table. Then as long as the center of the penny 'lands' no further than 3/8 of an inch from an edge of the table, it hits the table. Then there's a much larger square the penny can 'land' on -- it would measure 1 + 3/8 + 3/8 inches on each side, or 7/4 inches on each side, for a total area of 49/16 square inches. Then the probability of 'winning' is (1/16)/(49/16) = 1/49.
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In a common carnival game, a player tosses a penny from a distance ont 17 Mar 2021, 05:28
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Fdambro294
Beautifully done.

Although you lost me a bit on the end lol.

I think you mean the center of the penny has to just hit the edge.....otherwise if it goes too far, it won’t be balanced and it will fall off?
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There's a problem with the wording of the question, when it says: "If the penny... falls entirely inside a square". When it says "a square", that suggests there's more than one square the penny could land within, but the start of the question describes just one square table. So I was left to guess how to interpret it. If there is just one small square table, then when the question tells us at the end "the penny lands on the table", that could mean "the center of the penny lands on the table" (which is what would need to happen for the penny to not fall off, assuming the penny lands and then doesn't move at all), and in that case the answer is 1/16. But if "the penny lands on the table" simply means "some part of the penny, even if just a corner of it, hits the table", then the answer is 1/49.

I wonder if the question means instead to describe a very large table which is 'checkered' with 1 inch by 1 inch squares. Then the answer is 1/16, provided we know the penny doesn't land around the edges of the table.
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In a common carnival game, a player tosses a penny from a distance ont 17 Mar 2021, 18:40
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Ah, now I gotcha. Thank you 🙏



IanStewart
Fdambro294
Beautifully done.

Although you lost me a bit on the end lol.

I think you mean the center of the penny has to just hit the edge.....otherwise if it goes too far, it won’t be balanced and it will fall off?

There's a problem with the wording of the question, when it says: "If the penny... falls entirely inside a square". When it says "a square", that suggests there's more than one square the penny could land within, but the start of the question describes just one square table. So I was left to guess how to interpret it. If there is just one small square table, then when the question tells us at the end "the penny lands on the table", that could mean "the center of the penny lands on the table" (which is what would need to happen for the penny to not fall off, assuming the penny lands and then doesn't move at all), and in that case the answer is 1/16. But if "the penny lands on the table" simply means "some part of the penny, even if just a corner of it, hits the table", then the answer is 1/49.

I wonder if the question means instead to describe a very large table which is 'checkered' with 1 inch by 1 inch squares. Then the answer is 1/16, provided we know the penny doesn't land around the edges of the table.
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In a common carnival game, a player tosses a penny from a distance ont 18 Apr 2024, 00:02
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Here's a diagram explaining the solution above (not drawn to scale). Essentially, because the coin has to always be inside the square, that means that the center of the coin must be at least 3/8 inches away from all sides at all times. This means that we can essentially construct a rectangle within the square of all the possible places where the center of the coin can go (NOT DRAWN TO SCALE):­



As shown above, we essentially constructed a 1/4 * 1/4 square inside. Finding the area of it, we get 1/16, C.­
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