5.
An examination paper consists of 8 questions, of which one is on geometric distributions and one is on binomial distributions.
- If the 8 questions are arranged in a random order, find the probability that the question on geometric distributions is next to the question on binomial distributions.
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[2]
Four of the questions, including the one on geometric distributions, are worth 7 marks each, and the remaining four questions, including the one on binomial distributions, are worth 9 marks each. The 7 -mark questions are the first four questions on the paper, but are arranged in random order. The 9 -mark questions are the last four questions, but are arranged in random order. Find the probability that - the questions on geometric distributions and on binomial distributions are next to one another,
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[2] - the questions on geometric distributions and on binomial distributions are separated by at least 2 other questions.
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[3]
\section*{6.}
The random variable \(X\) represents the weight in kg of a randomly selected male dog of a particular breed. \(X\) is Normally distributed with mean 30.7 and standard deviation 3.5.
i) Find the \(90 ^ { \text {th } }\) percentile for the weights of these dogs.
ii) Five of these dogs are chosen at random. Find the probability that exactly four of them weighs at least 30 kg .
The weights of females of the same breed of dog are Normally distributed with mean 26.8 kg .
iii) Given that \(5 \%\) of female dogs of this breed weigh more than 30 kg , find the standard deviation of their weights.
iv) Sketch the distributions of the weights of male and female dogs of this breed on a single diagram.