Binomial Distribution: Average & Probability of ≥1 Success

In summary, the binomial distribution with probability k for success can be represented by <> = Nk. This means that the distribution function is peaked around 1 when <> = 1. When analyzing the probability of having at least one success in N tries, the approximation P(≥1) = Nk can be used, but only when Nk is sufficiently small. This approximation does not hold for Nk>1 and may not hold for small N. The binomial theorem can also be used to find the probability of at least one success, but the first term must dominate, which occurs when Nk is small.
  • #1
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The average if the binomial distribution with probability k for succes is simply:

<> = Nk

So this means that if <> = 1 the distribution function must be peaked around 1. In general when is it a good approximation (i.e. when is the function peaked sufficiently narrow) to say that the probability in N tries to have one or more succeses is simply:

P(≥1) = Nk

this obviously does not hold for Nk>1 but on the other hand I don't expect it to hold for small N. So my guess is when Nk is sufficiently small. Is that correct?
 
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  • #2
P(at least one success) = 1 - P(all failure) = 1 - (1-k)N.
You should be able to analyze it.
 
  • #3
Right so using the binomail theorem you find:

P(at least one success) = Nk - K(N,2)k2 + K(N,3)k3 - K(N,4)k4 + ...

So the question is when the first term dominates. I am guessing for sufficiently small k?
 
  • #4
You need Nk small, not just k. In your expression you seem to have extraneous K (capital k) from the second term on.
 
  • #5


Your understanding is partially correct. The binomial distribution with probability k for success has an average of Nk. This means that if you were to repeat the experiment many times, on average, you would expect to have Nk successes. However, the distribution is not necessarily peaked around this average value. It can be spread out depending on the value of N and k.

To answer your question, it is not accurate to say that the probability of having one or more successes in N tries is simply Nk. This only holds if Nk is less than or equal to 1. If Nk is greater than 1, then the probability of having at least one success is actually 1. This is because having one or more successes is guaranteed if Nk is greater than 1.

In general, the probability of having at least one success in N tries can be calculated using the binomial distribution formula: P(≥1) = 1 - (1-k)^N. This formula takes into account the fact that there can be more than one success in N tries.

To determine when the binomial distribution is a good approximation for the probability of having one or more successes, you can look at the shape of the distribution. If the distribution is relatively narrow and peaked around the average value of Nk, then it is a good approximation. However, if the distribution is spread out and not peaked around Nk, then it may not be a good approximation. This is dependent on the values of N and k, so there is no specific cutoff for when it is a good approximation. It is always best to use the formula for calculating the probability of at least one success to get an accurate result.
 

FAQ: Binomial Distribution: Average & Probability of ≥1 Success

What is a binomial distribution?

A binomial distribution is a probability distribution that describes the likelihood of obtaining a certain number of successes in a fixed number of trials, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant throughout all trials.

What is the formula for calculating the average of a binomial distribution?

The formula for calculating the average of a binomial distribution is μ = n * p, where μ is the average, n is the number of trials, and p is the probability of success for each trial.

How do you calculate the probability of at least one success in a binomial distribution?

The probability of at least one success in a binomial distribution can be calculated by subtracting the probability of zero successes from 1. This can be represented as P(x ≥ 1) = 1 - P(x = 0).

What is the relationship between the average and the probability of at least one success in a binomial distribution?

The average of a binomial distribution and the probability of at least one success are directly related. As the average increases, the probability of at least one success also increases. This is because a higher average means a higher number of trials, increasing the chances of at least one success occurring.

How can binomial distributions be applied in real life?

Binomial distributions can be used to model a variety of real-life scenarios, such as predicting the success rate of a new product launch, estimating the chances of winning a game or competition, or calculating the probability of a certain number of accidents occurring in a given time period.

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