5.01a Permutations and combinations: evaluate probabilities

Miguel has six numbered tiles, labelled 2, 2, 3, 3, 4, 4. He selects two tiles at random, without replacement. The variable \(M\) denotes the sum of the numbers on the two tiles.

1. Show that \(P(M = 6) = \frac{1}{3}\) [2]

The table shows the probability distribution of \(M\)

\(m\)	4	5	6	7	8
\(P(M = m)\)	\(\frac{1}{15}\)	\(\frac{4}{15}\)	\(\frac{1}{3}\)	\(\frac{4}{15}\)	\(\frac{1}{15}\)

Miguel returns the two tiles to the collection. Now Sofia selects two tiles at random from the six tiles, without replacement. The variable \(S\) denotes the sum of the numbers on the two tiles that Sofia selects.

1. Find \(P(M = S)\) [3]
2. Find \(P(S = 7 | M = S)\) [4]

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. [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

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

26 cards are each labelled with a different letter of the alphabet, A to Z. The letters A, E, I, O and U are vowels.

1. Five cards are selected at random without replacement. Determine the probability that the letters on at least three of the cards are vowels. [4]
2. All 26 cards are arranged in a line, in random order.
   1. Show that the probability that all the vowels are next to one another is \(\frac{1}{2990}\). [3]
   2. Determine the probability that three of the vowels are next to each other, and the other two vowels are next to each other, but the five vowels are not all next to each other. [4]

Bob has been given a pile of five letters addressed to five different people. He has also been given a pile of five envelopes addressed to the same five people. Bob puts one letter in each envelope at random.

1. How many different ways are there to pair the letters with the envelopes? [1]
2. Find the number of arrangements with exactly three letters in the correct envelopes. [2]
3. 1. Show that there are two derangements of the three symbols A, B and C. [1]
   2. Hence find the number of arrangements with exactly two letters in the correct envelopes. [1]

Let \(D_n\) represent the number of derangements of \(n\) symbols.

1. Explain why \(D_n = (n-1) \times (D_{n-1} + D_{n-2})\). [2]
2. Find the number of ways in which all five letters are in the wrong envelopes. [2]

The letters of the word 'SEPARATE' are to be rearranged. Find the probability that, in a randomly chosen rearrangement, the two letters 'A' are not next to each other. [4]