Probability of flipping a biased coin k times

• mcnkevin
In summary, the conversation is discussing the probability that a coin with a biased probability of heads and tails will result in more tails than heads when flipped k times. There is confusion over the specific question and the correct probability calculation, but it involves considering all possible outcomes where tails is flipped the desired number of times.
mcnkevin

Homework Statement

I have an unfair coin that comes ups heads 2/3 of the time and tails 1/3 of the time. If i flip this coin k times, what is probability that tails came up more then heads. Assume k is odd. So basically, what is the probability that the number of tails > k/2?

The Attempt at a Solution

I got a mental block right now. Obviously the probability will depend on k, and the larger the the lower the probability. Is it correct if I say the probability is (1/3)^(k/2+1)?

Thanks

Last edited:
Sorry if this is a stupid question, but how can tails come up more than the times you flip the coin?

If you flip it k times, it can't come up more than k times.

it says k/2 times not k.

mcnkevin said:
If i flip this coin k times, what is probability that tails came up more then k.

0

Mezzlegasm was right, something is wrong here. Should it be

If i flip this coin k times, what is probability that tails came up more then k/2?

Oh right, yes. I think i actually meant to say in that sentance, what is the probability that it comes up more then heads. Later i mention k/2. Sorry.

mcnkevin said:
I have an unfair coin that comes ups heads 2/3 of the time and tails 1/3 of the time. If i flip this coin k times, what is probability that tails came up more then k.
Editorial note: This should say "what is probability that tails came up more then heads", not k. This is what led to Mezzlegasm's post.

Is it correct if I say the probability is (1/3)^(k/2+1)?
No. This isn't even true for the trivial case, k=1, in which the probability is obviously 1/3. What is the probability for k=3? What kind of distribution is this?

mcnkevin said:
Oh right, yes. I think i actually meant to say in that sentance, what is the probability that it comes up more then heads. Later i mention k/2. Sorry.

Yeah, I missed the latter part because I was confused with the former.

Anyhow, I'm pretty sure that it is (1/3)^(k/2). I always had a hard time with counting principles and probability, so I need someone to confirm that for me as well.

EDIT: alright, I've just seen the above post. My answer didn't seem to make sense because it gave a different probability with more coin tosses, which shouldn't happen (I don't think). I couldn't find a single way to make it work, and the probability that it comes up more than heads seems to be a more appropriate question. Regardless, my answer is flawed.

Not sure what you mean by what kind of distribution? Basically, each toss is independant and each toss you have 1/3 of a chance of flipping tails and 2/3 a chance flipping heads. If you make k flips, what's the probability tails was flipped more times then heads.

For k = 3 , well you have to flip 2 tails in 3 tries.

So the possibilities are
All three tails = (1/3)(1/3)(1/3)
First two tails = (1/3)(1/3)(2/3)
First and last tails (1/3)(2/3)(1/3)
Second and last tails = (2/3)(1/3)(1/3)

So basically, (1/3)^3 + (3*((1/3)^2)*(2/3))

Correct? It certainly may be very wrong.. but if it is correct, how could i generalize this to k?

mcnkevin said:
Not sure what you mean by what kind of distribution? Basically, each toss is independant and each toss you have 1/3 of a chance of flipping tails and 2/3 a chance flipping heads. If you make k flips, what's the probability tails was flipped more times then heads.

For k = 3 , well you have to flip 2 tails in 3 tries.

So the possibilities are
All three tails = (1/3)(1/3)(1/3)
First two tails = (1/3)(1/3)(2/3)
First and last tails (1/3)(2/3)(1/3)
Second and last tails = (2/3)(1/3)(1/3)

So basically, (1/3)^3 + (3(1/3)^2(2/3))

Correct? It certainly may be very wrong.. but if it is correct, how could i generalize this to k?

Ah alright, this is starting to feel familiar now.

Like you said there are different possibilities, so you have to add the different probabilities that have tails at the desired amount. I think this is in the right direction, someone correct me if I'm wrong. I'm not sure where to go now, but I believe it starts with

(1/3)^k +

then you would have to add the different possible probabilities when considering how many times you can flip heads and still have desirable results. I just don't know how to do this.

1. What is the probability of getting a head when flipping a biased coin?

The probability of getting a head when flipping a biased coin is dependent on the bias of the coin. If the coin is biased to favor heads, then the probability will be higher than 50%. If the coin is fair, then the probability is 50%.

2. How does the number of flips affect the probability of getting a certain outcome?

The more times a biased coin is flipped, the closer the results will be to the expected probability. For example, if a coin is biased with a 70% chance of landing on heads, flipping it 100 times will result in approximately 70 heads and 30 tails.

3. Can the probability of getting a certain outcome change over time?

No, the probability of getting a certain outcome when flipping a biased coin will remain the same as long as the bias of the coin remains the same. The outcomes of previous flips do not affect the probability of future flips.

4. How can you determine if a coin is biased?

To determine if a coin is biased, you can flip it multiple times and record the results. If the results show a significant difference from the expected probability, then the coin is likely biased. Alternatively, you can use statistical methods such as hypothesis testing to determine if the coin is biased.

5. Is there a way to manipulate the probability of getting a certain outcome when flipping a biased coin?

Yes, the bias of a coin can be changed by altering its physical properties, such as its weight distribution or surface texture. However, manipulating a coin's bias is considered cheating and is not a reliable way to change the probability of getting a certain outcome when flipping a coin.

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