Toss coin probabilities (Bino vs Gauss approx)

In summary, we discussed the probability of getting between 3 and 6 heads after tossing a coin 10 times, and calculated the binomial probability and Gaussian approximation for this scenario. We found that the two probabilities were significantly different, with the Gaussian approximation overestimating the binomial probability by about 14%. This is likely due to the fact that the rule of thumb for using the Gaussian approximation, np and n(1-p) both being greater than 5, was not met. We also discussed the need for a continuity correction in the Gaussian approximation.
  • #1
ChrisVer
Gold Member
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Hi, I calculated the probability to this scenario:
getting between 3 and 6 Heads after tossing a coin for n=10 trials...

The binomial probability for this is:
[itex]P(3 \le k \le 6) = \sum_{k=3}^6 Bi(k;p,n)= \sum_{k=3}^6 \frac{n!}{k!(n-k)!} p^k (1-p)^{n-k} =\sum_{k=3}^6 \frac{10!}{k!(10-k)!} (0.5)^{10} = 0.7734375 [/itex]

Whereas for the Gaussian approximation to the binomial:
[itex] G(x) = \frac{1}{\sqrt{2 \pi n p (1-p) }} \exp\Big( - \frac{ (x-np)^2}{2np(1-p)} \Big)[/itex]

I get:
[itex]P(3 \le x \le 6) = \int_3^6 dx~ G(x) = ... = 0.633505 [/itex]

My question is why are the two probabilities so off from each other? Is it because [itex]n[/itex] cannot be considered as "large" ? I think when extracting the Gauss approximation, the additional higher orders go as [itex]\frac{1}{n}= 0.1[/itex] but that's for the gaussian distribution ##G## and not the integral of it.
Any feedback?
 
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  • #3
Simon Bridge said:
n=10 p=0.5 gives [np, n(1-p)] = [5,5] and you really want them both bigger than 5.

So the difference is mainly due to the fact that [itex]np= n(1-p)=5[/itex] is not large enoguh? So Gauss can go a bit off in its predictions (here ~10%)...

The strange thing is that this is the only rule the approximation break, whereas the other two rules give:
[itex] \Big| \frac{1}{\sqrt{n}} \Big( \sqrt{\frac{q}{p}} -\sqrt{\frac{p}{q}} \Big) \Big| = 0 <0.3 [/itex]
and
[itex] np \pm 3 \sqrt{npq} \in [0.26 , 9.74 ] [/itex] which is in [0,n=10]...

Simon Bridge said:
There are online calculators that will help you check your answers.
i.e. http://stattrek.com/online-calculator/normal.aspx

For the calculations I try to build my own source codes just for a practice :smile: (or check with the formulas in wolframalpha), and so far my numerical methods work fine.
 
  • #4
You are forgetting the continuity correction: $$ P(3 \le x \le 6) \approx \int_{2.5}^{6.5} G(x)~dx \approx 0.7717 $$
 

Related to Toss coin probabilities (Bino vs Gauss approx)

1. What is the difference between Bino and Gauss approximations in toss coin probabilities?

The Bino approximation, or binomial distribution, is a discrete probability distribution that models the number of successes in a series of independent trials with two possible outcomes (such as heads or tails in a coin toss). The Gauss approximation, or normal distribution, is a continuous probability distribution that can be used to approximate the binomial distribution when the number of trials is large. In other words, the Bino approximation is more accurate for a small number of trials, while the Gauss approximation is more accurate for a large number of trials.

2. How do Bino and Gauss approximations calculate the probability of getting a certain number of heads in a given number of coin tosses?

The Bino approximation uses the formula P(x) = nCx * p^x * (1-p)^(n-x) where n is the number of trials, x is the number of successes, and p is the probability of success on a single trial. The Gauss approximation uses the formula P(x) = e^(-(x-np)^2/(2np(1-p))) / sqrt(2πnp(1-p)), which approximates the Bino formula when n (number of trials) is large.

3. Which approximation is more accurate for calculating the probabilities of a coin toss when the number of trials is small?

The Bino approximation is more accurate for small numbers of trials. For example, if you are tossing a coin 10 times, the Bino approximation will be more accurate in calculating the probability of getting exactly 5 heads. However, as the number of trials increases, the Gauss approximation becomes more accurate.

4. Can Bino and Gauss approximations be used for any coin toss game, or are they specifically for a fair coin?

Bino and Gauss approximations can be used for any coin toss game, as long as the probability of getting a certain outcome (such as heads or tails) remains constant for each trial. This means that the coin must be fair, with an equal chance of landing on either side.

5. Are there any limitations to using Bino and Gauss approximations for calculating coin toss probabilities?

One limitation of using these approximations is that they assume the coin tosses are independent, meaning that the outcome of one toss does not affect the outcome of the next toss. In reality, this may not always be the case, especially if the coin is biased or there are external factors that could influence the outcome. Additionally, these approximations are based on theoretical models and may not always accurately predict the actual outcomes in real-life scenarios.

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