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A card is drawn at random from an ordinary deck of 52 playing cards. If we are told that card is heart, then information, in bits, is(Answer up to the nearest integer)
Correct answer is '2'. Can you explain this answer?
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A card is drawn at random from an ordinary deck of 52 playing cards. ...
Understanding the problem
In this problem, we are given a standard deck of 52 playing cards and we need to determine the amount of information, in bits, gained when we are told that the drawn card is a heart.
Explanation
To understand why the answer is 2 bits, let's first understand the concept of information theory. Information theory is a branch of mathematics that deals with quantifying information and measuring the amount of uncertainty or surprise in an event.
In this case, we have a deck of 52 playing cards, and each card has an equal probability of being drawn. Without any prior information, the probability of drawing a specific card is 1/52.
When we are told that the drawn card is a heart, we gain information that reduces the uncertainty. Now, we know that the card belongs to a specific suit (hearts) out of the four possible suits (hearts, diamonds, clubs, and spades). The probability of drawing a heart card is 1/4.
Calculating the amount of information
To calculate the amount of information gained, we can use the formula:
I = -log2(p)
where I is the information gained in bits and p is the probability of the event.
In this case, the probability of drawing a heart card is 1/4. Plugging this value into the formula:
I = -log2(1/4) = -log2(1) + log2(4) = 0 + 2 = 2 bits
Therefore, when we are told that the drawn card is a heart, we gain 2 bits of information.
Summary
The answer to the problem is 2 bits because when we are told that the drawn card is a heart, we gain information that reduces the uncertainty and narrows down the possibilities from 52 cards to just the 13 heart cards. This reduction in uncertainty corresponds to 2 bits of information.
In this problem, we are given a standard deck of 52 playing cards and we need to determine the amount of information, in bits, gained when we are told that the drawn card is a heart.
Explanation
To understand why the answer is 2 bits, let's first understand the concept of information theory. Information theory is a branch of mathematics that deals with quantifying information and measuring the amount of uncertainty or surprise in an event.
In this case, we have a deck of 52 playing cards, and each card has an equal probability of being drawn. Without any prior information, the probability of drawing a specific card is 1/52.
When we are told that the drawn card is a heart, we gain information that reduces the uncertainty. Now, we know that the card belongs to a specific suit (hearts) out of the four possible suits (hearts, diamonds, clubs, and spades). The probability of drawing a heart card is 1/4.
Calculating the amount of information
To calculate the amount of information gained, we can use the formula:
I = -log2(p)
where I is the information gained in bits and p is the probability of the event.
In this case, the probability of drawing a heart card is 1/4. Plugging this value into the formula:
I = -log2(1/4) = -log2(1) + log2(4) = 0 + 2 = 2 bits
Therefore, when we are told that the drawn card is a heart, we gain 2 bits of information.
Summary
The answer to the problem is 2 bits because when we are told that the drawn card is a heart, we gain information that reduces the uncertainty and narrows down the possibilities from 52 cards to just the 13 heart cards. This reduction in uncertainty corresponds to 2 bits of information.
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Community Answer
A card is drawn at random from an ordinary deck of 52 playing cards. ...
P (heart) = 13/52 =1/4
I = log24 = 2 bits
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A card is drawn at random from an ordinary deck of 52 playing cards. If we are told that card is heart, then information, in bits, is(Answer up to the nearest integer)Correct answer is '2'. Can you explain this answer?
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