Specific items together

8 An examination paper consists of 8 questions, of which one is on geometric distributions and one is on binomial distributions.

1. 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. 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
2. the questions on geometric distributions and on binomial distributions are next to one another,
3. the questions on geometric distributions and on binomial distributions are separated by at least 2 other questions.

4 The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.

1. Find the probability that all the men are next to each other.
2. Find the probability that no two men are next to one another.

6 A bag contains 6 identical blue counters and 5 identical yellow counters.

1. Three counters are selected at random, without replacement. Find the probability that at least two of the counters are blue. All 11 counters are now arranged in a row in a random order.
2. Find the probability that all the yellow counters are next to each other.
3. Find the probability that no yellow counter is next to another yellow counter.
4. Find the probability that the counters are arranged in such a way that both of the following conditions hold.
   • Exactly three of the yellow counters are next to one another.
   • Neither of the other two yellow counters is next to a yellow counter.
   • Explain whether the answer to part (d) would be different if the yellow counters were numbered \(1,2,3,4\) and 5 , so that they are not identical.

11. The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.

1. Find the probability that all the men are next to each other.
2. Find the probability that no two men are next to one another.

3. The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.

1. Find the probability that all the men are next to each other.
2. Find the probability that no two men are next to one another.
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6. (a) The members of a team stand in a random order in a straight line for a photograph. There are four men and six women. Find the probability that all the men are next to each other.
(b) Find the probability that no two men are next to one another.
[0pt] [BLANK PAGE]

5. An examination paper consists of 8 questions, of which one is on geometric distributions and one is on binomial distributions.

1. 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.
   [0pt] [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
2. the questions on geometric distributions and on binomial distributions are next to one another,
   [0pt] [2]
3. the questions on geometric distributions and on binomial distributions are separated by at least 2 other questions.
   [0pt] [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.

2 Each letter of the words NEW COURSE is written on a card (including one blank card, representing the space between the words), so that there are 10 cards altogether.

1. All 10 cards are arranged in a random order in a straight line. Find the probability that the two cards containing an E are next to each other.
2. 4 cards are chosen at random. Find the probability that at least three consonants ( \(\mathrm { N } , \mathrm { W } , \mathrm { C } , \mathrm { R } , \mathrm { S }\) ) are on the cards chosen.