Haven't looked at statistics in a while, and came across the following problem:
It is estimated that approximately 20% of marketing calls result in a sale. Assume that 10 out of 12 marketing calls on a day resulted in a sale. What is the probability that the first call made is one of the 10 that resulted in a sale?
As each call is independent, I am incorrect in thinking that the answer is simply 10/12? I just cant figure out how to prove it mathematically. I assume that it has something to do with proving independence of the choosing of the 1st from the assumption that 10 of the 12 calls were successes, but im just not sure where to start.
Thanks in advance!
the result is independent of the prior probability of $20\%$. In fact any of the 66 combination with 10 success among 12 calls are equiprobable.
Thus the result is only a % of the favourable combinations on the possible combinations
$$\frac{55}{66}=\frac{5}{6}$$
this is the sampling space
Of course it is not necessary to write down all the sample space to derive that the favourable events are 55. It is enough to observe that, if the first element in the succession is "1" there are $\binom{11}{9}=\binom{11}{2}=55$ different ways to combine the remaining 11 elements with 9 successes