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September 20th, 2019, 08:15 AM   #1
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Probability When Four People Flip Coins

One at a time, four people flip coins. Each person flips until they get one wrong, and their score is how many they get correct. What is the probability that the four people combine to score at least 5? When the game is done, everybody will have been wrong once for a total of being wrong four times. Is the answer the same as the probability of one person scoring at least 5 if he or she flipped until he or she was wrong four times?
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September 20th, 2019, 10:12 AM   #2
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If I understand your question the underlying experiment doesn't matter.

As long as being a different person has no effect on the outcome of the experiment
it doesn't matter if your samples come from 4 people or 1 person repeating the experiment 4 times.
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September 20th, 2019, 10:40 AM   #3
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Assuming the probability of being right is 50 %, the odds of one person scoring 0 is 1/2. The odds of scoring 1 (first right, then wrong), is 1/4. The odds of scoring S is 1/2^(S-1).

To answer your question, it is easier to calculate the probability of four people scoring 4 or less.
Let $T = S_1+S_2+S_3+S_4$ be the combined score.
T = 0 means all four people scored 0. (1/2)*(1/2)*(1/2)*(1/2) = 1/16
T = 1 means the scores were 1,0,0,0 (odds 1/4*1/2*1/2*1/2 = 1/32) or 0,1,0,0 (odds 1/32) or 0,0,1,0 (odds 1/32) or 0,0,0,1 (odds 1/32). Odds of T = 1: 4*(1/32) = 1/8
Keep going for T = 2, T = 3, and T = 4, then add them together.

If you know how to calculate numbers of combinations, you can simplify the counting somewhat.
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September 20th, 2019, 04:47 PM   #4
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What's wrong with this?

For T = 2, there are ten possibilities:

2-0-0-0: (1/8)(1/2)(1/2)(1/2) = 1/64
0-2-0-0: (1/2)(1/8)(1/2)(1/2) = 1/64
0-0-2-0: (1/2)(1/2)(1/8)(1/2) = 1/64
0-0-0-2: (1/2)(1/2)(1/2)(1/8) = 1/64
Subtotal: 4/64

1-1-0-0: (1/4)(1/4)(1/2)(1/2) = 1/64
1-0-1-0: (1/4)(1/2)(1/4)(1/2) = 1/64
1-0-0-1: (1/4)(1/2)(1/2)(1/4) = 1/64
0-1-1-0: (1/2)(1/4)(1/4)(1/2) = 1/64
0-0-1-1: (1/2)(1/2)(1/4)(1/4) = 1/64
0-1-0-1: (1/2)(1/4)(1/2)(1/4) = 1/64
Subtotal: 6/64

The sum is 10/64, which is 5/32.

For T = 3, there are 24 possibilities:

3-0-0-0: (1/16)(1/2)(1/2)(1/2) = 1/128
0-3-0-0: (1/16)(1/2)(1/2)(1/2) = 1/128
0-0-3-0: (1/16)(1/2)(1/2)(1/2) = 1/128
0-0-0-3: (1/16)(1/2)(1/2)(1/2) = 1/128
Subtotal: 4/128

2-1-0-0: (1/8)(1/4)(1/2)(1/2) = 1/128
2-0-1-0: (1/8)(1/2)(1/4)(1/2) = 1/128
2-0-0-1: (1/8)(1/2)(1/2)(1/4) = 1/128
1-2-0-0: (1/4)(1/8)(1/2)(1/2) = 1/128
1-0-2-0: (1/4)(1/2)(1/8)(1/2) = 1/128
1-0-0-2: (1/4)(1/2)(1/2)(1/8) = 1/128
0-2-1-0: (1/2)(1/8)(1/4)(1/2) = 1/128
0-2-0-1: (1/2)(1/8)(1/2)(1/4) = 1/128
0-0-2-1: (1/2)(1/2)(1/8)(1/4) = 1/128
0-0-1-2: (1/2)(1/2)(1/4)(1/8) = 1/128
0-1-2-0: (1/2)(1/4)(1/8)(1/2) = 1/128
0-1-0-2: (1/2)(1/4)(1/2)(1/8) = 1/128
Subtotal: 12/128

1-1-1-0: (1/4)(1/4)(1/4)(1/2) = 1/128
1-1-0-1: (1/4)(1/4)(1/2)(1/4) = 1/128
1-0-1-1: (1/4)(1/2)(1/4)(1/4) = 1/128
0-1-1-1: (1/2)(1/4)(1/4)(1/4) = 1/128
Subtotal: 4/128

The total is 20/128 = 5/32

The total of 0, 1, 2, and 3 is 1/16 + 1/8 + 5/32 + 5/32 = 16/32 = 1/2. That looks wrong because the expected value is 4, and I don't think the probability of being less than the expected value would be 1/2. For example, if you flip an even amount of coins so that the expected value is a whole number, the probability of more heads than the expected value and of fewer heads than the expected value will both be under 1/2. This is different because it's not based on a fixed number of trials, but I'd still be surprised if the total of 0, 1, 2, and 3 is 1/2. If what I'm doing is correct, I can do the probability for 4 on my own, but I want to know if I'm correct.
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September 21st, 2019, 02:47 PM   #5
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No, you were right. With 4 people playing, there is a 50 % chance of getting a score of 4 or higher. Validated against 10^6 simulations.

4 is not the most likely score, however.
Attached Images
File Type: jpg CoinFlipScore_N04.jpg (20.3 KB, 1 views)
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September 21st, 2019, 03:18 PM   #6
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Additional fun facts (which would probably be obvious to mathematicians if they expressed the probabilities in equation form):
* If N people are playing, there is a 50 % chance of the total score being N or higher (not surprising -- 50 % chance of getting a score of zero).
* If N people are playing, the most probable total score is tied between N-1 and N-2 (except if N = 1, then the most probable score is N-1 = 0).


Now a question for the statisticians: As $N \rightarrow \infty$ (with the scores appropriately scaled), does this distribution approach a Poisson distribution, a lognormal distribution, or something else?
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September 21st, 2019, 05:25 PM   #7
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Quote:
Originally Posted by EvanJ View Post
One at a time, four people flip coins. Each person flips until they get one wrong, and their score is how many they get correct. What is the probability that the four people combine to score at least 5? When the game is done, everybody will have been wrong once for a total of being wrong four times. Is the answer the same as the probability of one person scoring at least 5 if he or she flipped until he or she was wrong four times?
Let's make a few changes to question:
Each person flips until they get Tails, and every time one gets Heads earns a 1 point.

Technically, there is no difference between 4 people tossing a coin, each or one person tossing 4 coins, given he (or she) can toss coins which came Heads, again.

# What is the probability of getting 0 points?
$${[Pr(Tails)]}^4=1/16=0.0625$$

# What is the probability of getting 1 point?
You get 1 Heads on the first round and 0 on the second round.
$$Pr( \text{1 Heads && 3 Tails}) \times Pr( \text{1 Tails}) = \frac{4}{16} \times \frac{1}{2} = 1/8$$

# What is the probability of getting 2 points?
Either you get 2 points on the first round and 0 on the second OR get 1 point on the first and second round and 0 on the third.

$$[Pr( \text{2 points on first round}) \times Pr( \text{0 points on second round})] + [Pr( \text{1 point on first round}) \times Pr( \text{1 points on second round}) \times Pr( \text{0 points on third round})] = \frac{6}{16} \times \frac{1}{4} + \frac{4}{16} \times \frac{1}{2} \times \frac{1}{2}=\frac{5}{32}$$


Do it for 3 points and 4 points.
$$Pr( \text{At least 5 points}) = 1- Pr( \text{At most 4 points})$$
$$ Pr( \text{At most 4 points}) = Pr( \text{1 point}) + Pr( \text{2 points})+ Pr( \text{3 points}) + Pr( \text{4 points})$$

NOTE: You shouldn't use it as the solution for the problem. It is rather an explanation of how you can approach such problems.

Now for the "Is the answer the same as the probability of one person scoring at least 5 if he or she flipped until he or she was wrong four times?"
No, it is not.
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September 27th, 2019, 06:44 PM   #8
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Quote:
Originally Posted by tahirimanov19 View Post
Technically, there is no difference between 4 people tossing a coin, each or one person tossing 4 coins, given he (or she) can toss coins which came Heads, again.

Now for the "Is the answer the same as the probability of one person scoring at least 5 if he or she flipped until he or she was wrong four times?"
No, it is not.
These seem like opposite answers to the same question. Can you tell me the difference between the statements? If there is no difference between four people flipping until one failure each or one person flipping until four failures, there is no difference regardless of how many trials and successes there are.
romsek said the answer is the same.
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September 27th, 2019, 08:42 PM   #9
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I think EvanJ is right. Regardless who's flipping the coins, we're flipping coins until we get four "wrong" flips, and counting the "correct" ones. I don't see a difference in the probabilities.
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