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# Binomial distribution examples

* denotes question which is meant for discussion on that stats forum, so no worked solution is given

### Question DP003

(a) In a college, the percentage of students who are dating is 5 per cent. In a random sample of 10, find the chance that at least 2 students are dating.

*(b) Using a suitable approximation, find the probability of at least 2 students dating in a sample of 100.

Solution

(a) We have the data, sample size n = 10, and probability of dating is p = 0.05.

Let X denote the number of students who are dating. We are asked to find probability X is equal or more than 2.

This is the same as the probability of 1 minus the probability X is less than 2 i.e. 1 minus the probability of the opposite event.

Recognize that the variable X has all the features of a binomial distribution, so the probabilities will be calculated using the binomial pdf.Now to do the calc.

p(X ≥2) = 1 - p(X < 2) = 1 - p(X=0) - p(X=1) = 1 - (0.95)^{2} - 10(0.05)(0.95)^{9} = 0.086.

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