Is Flipping Coins Sequentially Different from Simultaneously for Probability?

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Flipping a coin four times and flipping four coins simultaneously yield the same probability for getting exactly two heads, as both scenarios involve independent events. The probability calculated for both questions is 6/16, assuming the goal is to find exactly two heads. However, if the first question were to ask for at least two heads instead of exactly two, the probability would differ. The independence of each coin flip is crucial in determining the outcomes. Understanding the distinction between "exactly" and "at least" is essential for accurate probability calculations.
Punkyc7
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Question 1:
A coin is flipped 4 times. Find the probability that it lands heads twice.

Question 2:
Four coins are flipped simultaneously. Find the probability that exactly two of the land on heads


For the first one I made an H and T chart and got 6/16.

The second one I am thinking its the same thing.

My question is how is the second question any different from the first?
 
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Punkyc7 said:
Question 1:
A coin is flipped 4 times. Find the probability that it lands heads twice.

Question 2:
Four coins are flipped simultaneously. Find the probability that exactly two of the land on heads


For the first one I made an H and T chart and got 6/16.

The second one I am thinking its the same thing.

My question is how is the second question any different from the first?

Different questions, but same answer. You could get a different answer if the events were dependent on each other. However, tossing the coin doesn't affect the future coin tosses and you can treat it as tossing four coins at once. The fact that these are independent events is key.

__________

I noticed something about question 1. Is it asking for the probablility of exactly 2 heads or at least 2 heads? If at least 2 heads, then you would get a different answer.
 
Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?

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