N2011 P2 Q7

When I try to contact (by telephone) any of my friends in the evening, I know that on average the probability that I succeed is \(0.7\). On one evening I attempt to contact a fixed number, \(n\), of different friends. If I do not succeed with a particular friend, I do not attempt to contact that friend again that evening. The number of friends whom I succeed in contacting is the random variable \(R\).

1. State, in the context of this question, two assumptions needed to model \(R\) by a binomial distribution.[2]
2. Explain why one of the assumptions stated in part (i) may not hold in this context.[1]

Assume now that these assumptions do in fact hold.

3. Given that \(n=8\), find the probability that \(R\) is at least 6.[1]
4. Given that \(n=40\), use an appropriate approximation to find \(\mathrm P(R<25)\). State the parameters of the distribution you use.[4]