A blind lecturer can cook and invites a few friends around for dinner. His wife

Question

A blind lecturer can cook and invites a few friends around for dinner. His wife is also blind and they actually manage to get the table set before the guests arrive . Cutlery poses no problem, but the place mats which will be used have pictures of NZ rivers on them. The place mats definitely look better to the guests if they are the right way up, but they are rectangular and cannot be distinguished as right way up or upside down by the hosts. Unfortunately , the only child in the house is only 4 months old and while he might be talented , he cannot tell his parents how to arrange the place mats. There are a total of 7 adults having dinner, so things might get a little hairy. The baby does't get a place mat
a)What distribution would best be used to model the no. of place mats that have been placed correctly? Explain your selection of distribution.
b)If your lecturer loves having friends around for dinner, and there are always 7 place mats put on the table, what is the long run proportion of times he will have at least 5 of the place mats up the right way?
c)What is the mean and variance of the distribution of the no. of correctly aligned place mats?

Explanation / Answer

The binomial distribution will work here because we have a fixed number of trials (7), the same probability of success on each one, and each trial is independent of the others. The binomial theorem is P(x=k)= n!/(n-k)! k! p^k (1-p)^(n-k) Here n=7, p=0.5, and k= 5,6,7 (at least 5) For 5 we have 7!/5!2! 0.5^5 0.5^2= 0.164 For 6 we have 0.055 For 7 we have 0.001 Total is 0.22 Mean is 7*0.5= 3.5 Variance is 7*0.5*0.5= 1.75

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