Heads occurs 15 times out of 20 - Is it a fair coin?

• phono
In summary: If you have a coin that is highly biased, then the probability math won't apply, but it is possible to detect that as well.In summary, the textbook states that in 96% of cases, the number of heads in 20 flips will fall between 6 and 14 inclusive. This can be proven using the Binomial Theorem and the probability of getting k heads from n flips of a fair coin, which is n choose k divided by 2^n. However, it is not possible to determine if a coin is fair or unfair based on only 20 flips. It is necessary to have a larger sample size to accurately assess the fairness of a coin. Additionally, consistent reflexes and timing can potentially influence the outcome
phono
I have to answer this question by using references to expectation and probability as well as any other relevant considerations. My textbook says that 96% of the time, the number of heads in 20 flips will be between 6 and 14 inclusive. Prove this.

Ok I can prove the answer in the textbook using Binomial Theorem.

Probability of k heads from n flips of a fair coin is n choose k = n!/((n - k)! k!) divided by total possibiliries = 2^n

Sp P(5) heads = (20 choose 5)/1048576

Sum P(n) from n = 6 to 14
(38760 + 77520 + 125970 + 167960 + 184756 + 167960 + 125970 + 77520 + 38760)/1048576

= 1005176/1048576 ~ 0.9586 ~ 96%

Edit: The question actually says "do you think the coin is fair?", sorry. Using references to expectation as well.

Thanks

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You can't prove it is a fair coin. A fair coin can produce 15 heads out of 20 tosses and so can an "unfair" coin. What do you mean by an "unfair" coin? Perhaps if the probability of getting a given result with a fair coin were very low, but that result did come up you could say that the coin was unfair. But you need to decide what the cut off is: 10%? 5%? 1%? Then calculate the probability with a fair coin and see if it is below that.

Note that even a fair coin can produce 20 heads out of 20- just with very low proabability.

OK, all the probability theorem and all the formula works for a large number of tries, you have to consider that you never can use probability formulas for a small number of tries (although we do this in problems! but mathematically and physically it is NOT correct!).

Probability of 1/2 for coins can be seen when you try it i.e. 100000 times not 20 times! this is the key point.

So you can never judge if the coin is fair or unfair in 20 times.

PS. unfair means if the coins is made correct and it has the same physics and shape and weight and ... in all it's place and one side is not heavier than the other side.

Good luck

Be aware that with consistent reflexes and timing in the coin-capture you can learn to influence the outcome of what looks like a casual coin-toss. Not relevant to the mathematical aspects of the question - just a heads-up about coin tosses in the real world.

Agreed, you can get relatively long strings of heads or tails even with an honest coin and random launches/catches, but increasing the sample size evens those out.

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1. Is it possible for a coin to land on heads 15 times out of 20 flips?

Yes, it is possible for a coin to land on heads 15 times out of 20 flips. While the probability of this occurrence is low, it is not impossible.

2. How do you determine if a coin is fair?

A fair coin is one that has an equal chance of landing on heads or tails. To determine if a coin is fair, it must be flipped numerous times and the results should be close to a 50/50 split between heads and tails.

3. What could cause a coin to land on heads more frequently?

There are several factors that could potentially cause a coin to land on heads more frequently, such as the weight distribution of the coin, the force used to flip the coin, or external factors such as wind or surface conditions.

4. Is it possible for a coin to never land on heads?

Technically, yes, it is possible for a coin to never land on heads. However, the probability of this occurring is extremely low and would require an infinite number of flips.

5. Can the results of a coin flip be influenced by human intervention?

In theory, yes, the results of a coin flip could be influenced by human intervention. For a fair coin, the results should be random and not influenced by external factors. However, if the coin is not fair or if there is significant force applied to the flip, it could potentially be influenced.

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