# Conceptual problem for binomial distribution

I'm reading this book (Implementation: How Great Expectations in Washington Are Dashed in Oakland. 1973) and they discuss how hard it is to gain agreement of actors.

They set up a conceptual problem to illustrate this.

Assume that in any chain of decisions, the contracting parties have an 80% probability of reaching an agreement to proceed. Now assume that there are 70 separate agreements that much be reached. What is the probability that all 70 agreements will be successful.

They report that with an 80% probability of agreement, the probability of success after 70 agreements is 0.000000125 and that the number of agreements that reduce the probability to below 50 percent is 4.

The thing is: they do not really explain how they arrive at these probabilities. So, that's why I'm turning to you.

It seems like what is going on here is an exercise in the binomial distribution. Given an 80% probability of a successful trial, what is the probability that 70 trials will return all successes?

To put this in the language of coin tosses: If we assume a coin has a probability of turning up heads of 80%, and we flip the coin 70 times, what is the probability that you would get heads 70 straight times?

Have I got this right?

Thanks for any insight. Thanks, Simon

I hope I've explained myself

• I think it's a fine question (+1). But it seems to me that this exercise amounts to just so much unfounded playing with arithmetic, because it is implausible that the outcomes of 70 separate agreements among contracting parties would be statistically independent. (It is almost as unbelievable that anyone could reliably estimate a probability of agreement.) Thus, perhaps we ought to understand this calculation as an example of the ridiculous results that can be obtained when one makes ridiculous assumptions.
– whuber
May 29, 2015 at 19:26

This seems to accord with the statement that the fourth agreement is what drives the probability of success below $.5$, since $.8^3 = .512$ and $.8^4=.4096$.
Their answer of $0.000000125$ after $70$ successes seems a bit off, since $.8^{70}$ is about $0.000000165$, but it is possible that they have made an error. If they didn't make an error, they are assuming something slightly different from the binomial distribution, but who knows what that might be!
• dbinom gives the probability density function, i.e. the probability of observing a given outcome given the underlying binomial distribution. The syntax relevant to your question is dbinom(x,size,prob) where x is the number of successes, size is the number of trials, and prob is the probability of an individual success. Since the probability of an individual success is $.8$, you want prob=.8 and since you're concerned with the probability of "all agreements are successful" you want x=size. So for three agreements, you want dbinom(3,3,.8). May 30, 2015 at 3:24