\begin{align*} S & = \{ \text{(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)} \} \end{align*}

First part

\begin{align*} S & = \{ F_1 , F_2, F_3, NF_1, NF_2, NF_3, NF_4 \} \\ \\ \text{P(pen is not faulty)} & = {4 \over 7} \end{align*}

Second part

\begin{align*} \text{P(Second pen is faulty)} & = {3 \over 6} \\ & = {1 \over 2} \end{align*}

(i)

\begin{align*} \text{P('S' is chosen)} & = {2 \over 11} \end{align*}

(ii)

\begin{align*} \text{P('P' or 'I' is chosen)} & = {1 + 3 \over 11} \\ & = {4 \over 11} \end{align*}

(iii)

\begin{align*} \text{P(a vowel is chosen)} & = {1 + 3 \over 11} \\ & = {4 \over 11} \end{align*}

(iv)

\begin{align*} \text{P(a consonant is chosen)} & = 1 - \text{P(a vowel is chosen)} \\ & = 1 - {4 \over 11} \\ & = {7 \over 11} \end{align*}

(a)

(b)(i)

\begin{align*} \text{Total number of possible outcomes} & = 3 \times 4 \\ & = 12 \\ \\ \text{P(Same number on both cards)} & = {2 \over 12} \\ & = {1 \over 6} \end{align*}

(b)(ii)

\begin{align*} \text{P(Different number on cards)} & = 1 - \text{P(Same number on both cards)} \\ & = 1 - {1 \over 6} \\ & = {5 \over 6} \end{align*}

(b)(iii)

\begin{align*} \text{P(Larger number is 3)} & = {3 \over 12} \\ & = {1 \over 4} \end{align*}

(a)

(b)

\begin{align*} \text{Number of possible outcomes} & = 6 \times 6 \\ & = 36 \end{align*}

(c)(i)

\begin{align*} \text{P(Sum of two numbers is 7)} & = {4 \over 36} \\ & = {1 \over 9} \end{align*}

(c)(ii) Prime numbers have two factors, 1 and itself

\begin{align*} \text{P(Sum of two numbers is a prime number)} & = {3 + 4 + 6 + 4 \over 36} \\ & = {17 \over 36} \end{align*}

(c)(iii)

\begin{align*} \text{P(Sum of two numbers is not a prime number)} & = 1 - \text{P(Sum of two numbers is a prime number)} \\ & = 1 - {17 \over 36} \\ & = {19 \over 36} \end{align*}

(c)(iv) Even numbers are divisible by 2

\begin{align*} \text{P(Sum of two numbers is an even number)} & = {1 + 3 + 5 + 5 + 3 + 1 \over 36} \\ & = {1 \over 2} \end{align*}

(c)(v)

\begin{align*} \text{P(Sum of two numbers is an odd number)} & = 1 - \text{P(Sum of two numbers is an even number)} \\ & = 1 - {1 \over 2} \\ & = {1 \over 2} \end{align*}

(d)

\begin{align*} \text{P(Sum of two numbers is 7)} & = {1 \over 9} \\ \\ \text{P(Sum of two numbers is 8)} & = {3 \over 36} \\ & = {1 \over 12} \\ \\ \therefore \text{Sum of 7 is more } & \text{likely to occur} \end{align*}

(a)

(b)(i) Prime numbers have two factors, 1 and itself

\begin{align*} \text{P(} x + y \text{ is a prime number)} & = {4 \over 9} \end{align*}

(b)(ii)

\begin{align*} \text{P(} x + y \text{ is greater than 12)} & = {6 \over 9} \\ & = {2 \over 3} \end{align*}

(b)(iii)

\begin{align*} \text{P(} x + y \text{ is greater than 12)} & = {8 \over 9} \end{align*}

(c)(i) Odd numbers are not divisible by 2

\begin{align*} \text{P(} xy \text{ is an odd number)} & = {2 \over 9} \end{align*}

(c)(ii)

\begin{align*} \text{P(} xy \text{ is an even number)} & = 1 - \text{P(} xy \text{ is an odd number)} \\ & = 1 - {2 \over 9} \\ & = {7 \over 9} \end{align*}

(c)(iii)

\begin{align*} \text{P(} xy \text{ is most 40)} & = {5 \over 9} \end{align*}

Tree diagram:

(i)

\begin{align*} \text{P(Three heads)} & = \text{P(HHH)} \\ & = {1 \over 8} \end{align*}

(ii)

\begin{align*} \text{P(Exactly two heads)} & = \text{P(HHT or HTH or THH)} \\ & = {3 \over 8} \end{align*}

(iii)

\begin{align*} \text{P(At least two heads)} & = \text{P(HHT or HTH or THH or HHH)} \\ & = {4 \over 8} \\ & = {1 \over 2} \end{align*}

Possible outcomes:

\begin{align*} S & = \{ \text{RB, RR, BB, BR, WB, WR} \} \end{align*}

(i)

\begin{align*} \text{P(Two marbles are of same colour)} & = {2 \over 6} \\ & = {1 \over 3} \end{align*}

(ii)

\begin{align*} \text{P(One blue marble & one red marble)} & = {2 \over 6} \\ & = {1 \over 3} \end{align*}

(iii)

\begin{align*} \text{P(Two marbles are of different colours)} & = 1- \text{P(Two marbles are of same colour)} \\ & = 1 - {1 \over 3} \\ & = {2 \over 3} \end{align*}

(a)

\begin{align*} S & = \{ 11, 12, 13, 21, 22, 23, 31, 32, 33 \} \end{align*}

(b)(i)

\begin{align*} \text{P(Number divisible by 3)} & = \text{P(12 or 21 or 33)} \\ & = {3 \over 9} \\ & = {1 \over 3} \end{align*}

(b)(ii)

\begin{align*} \text{P(Number is a perfect square)} & = {0 \over 9} \\ & = 0 \end{align*}

(b)(iii) A prime number only has two factors, 1 and itself

\begin{align*} \text{P(A prime number is formed)} & = \text{P(11 or 13 or 23 or 31)} \\ & = {4 \over 9} \end{align*}

(b)(iv) A composite number has more than two factors

\begin{align*} \text{P(A composite number is formed)} & = {5 \over 9} \end{align*}

Sample space:

\begin{align*} S & = \{ \text{ BBB, GGG, BBG, BGB, GBB, GGB, GBG, BGG } \} \end{align*}

(i)

\begin{align*} \text{P(Three grandsons)} & = \text{P(BBB)} \\ & = {1 \over 8} \end{align*}

(ii)

\begin{align*} \text{P(Two grandsons & one granddaughter)} & = \text{P(BBG or BGB or GBB)} \\ & = {3 \over 8} \end{align*}

(iii)

\begin{align*} \text{P(One grandsons & two granddaughters)} & = \text{P(BGG or GBG or GGB)} \\ & = {3 \over 8} \end{align*}

Note: There are 25 outcomes in total

(a)(i)

\begin{align*} \text{P(Sum is 6)} & = \text{P(1 + 5 or 5 + 1 or 2 + 4 or 4 + 2 or 3 + 3)} \\ & = {5 \over 25} \\ & = {1 \over 5} \end{align*}

(a)(ii)

\begin{align*} \text{P(Same number on both spinners)} & = \text{P(1, 1 or 2, 2 or 3, 3 or 4, 4 or 5, 5)} \\ & = {5 \over 25} \\ & = {1 \over 5} \end{align*}

(a)(iii)

\begin{align*} \text{P(Different number on the spinners)} & = 1 - \text{P(Same numbers on both spinners)} \\ & = 1 - {1 \over 5} \\ & = {4 \over 5} \end{align*}

(a)(iv) A prime number only has two factors, 1 and itself

\begin{align*} \text{P(Two different prime numbers)} & = \text{P(2, 3 or 3, 2 or 2, 5 or 5, 2 or 3, 5 or 5, 3)} \\ & = {6 \over 25} \end{align*}

(b)

\begin{align*} \text{P(First number less than second number)} & = \text{P(1, 2/3/4/5 or 2, 3/4/5 or 3, 4/5 or 4, 5)} \\ & = {4 + 3 + 2 + 1 \over 25} \\ & = {2 \over 5} \end{align*}

(a)

(b)(i)

\begin{align*} \text{P(Score is odd)} & = {3 \over 12} \\ & = {1 \over 4} \end{align*}

(b)(ii)

\begin{align*} \text{P(Score is odd)} & = {9 \over 12} \\ & = {3 \over 4} \end{align*}

(b)(iii) A prime number only has two factors, 1 and itself

\begin{align*} \text{P(Score is a prime number)} & = {2 + 1 + 1 \over 12} \\ & = {1 \over 3} \end{align*}

(b)(iv)

\begin{align*} \text{P(Score is less than or equals to 8)} & = {10 \over 12} \\ & = {5 \over 6} \end{align*}

(b)(v)

\begin{align*} \text{P(Score is a multiple of 3)} & = {4 \over 12} \\ & = {1 \over 3} \end{align*}

(a)

(b)(i)

\begin{align*} \text{Number of outcomes} & = 6 \times 6 \\ & = 36 \\ \\ \text{P(Difference is 1)} & = {10 \over 36} \\ & = {5 \over 18} \end{align*}

(b)(ii)

\begin{align*} \text{P(Difference is non-zero)} & = {36 - 6 \over 36} \\ & = {5 \over 6} \end{align*}

(b)(iii) In the possibility diagram, there are 3 odd numbers in each row

\begin{align*} \text{P(Difference is odd)} & = {18 \over 36} \\ & = {1 \over 2} \end{align*}

(b)(iv) A prime number only has two factors, 1 and itself

\begin{align*} \text{P(Difference is a prime number)} & = {3 + 2 + 3 + 3 + 2 + 3 \over 36} \\ & = {4 \over 9} \end{align*}

(b)(v)

\begin{align*} \text{P(Difference is a multiple of 3)} & = {2 + 2 + 2 + 2 + 2 + 2 \over 36} \\ & = {1 \over 3} \end{align*}

Possible outcomes in the form (First draw, Second draw):

(i)

\begin{align*} \text{P(Both balls are prime)} & = \text{P(2, 5 or 2, 7 or 5, 2 or 5, 7 or 7, 2 or 7,5)} \\ & = {6 \over 20} \\ & = {3 \over 10} \end{align*}

(ii)

\begin{align*} \text{P(Sum of two numbers is odd)} & = \text{P(1, 2 or 1, 4 or 2, 1 or 2, 5 or 2, 7 or 4, 1 or 4, 5 or 4, 7 or 5, 2 or 5, 4 or 7, 2 or 7, 4)} \\ & = {12 \over 20} \\ & = {3 \over 5} \end{align*}

(iii)

\begin{align*} \text{P(Product of two numbers is greater than 20)} & = \text{P(4, 7 or 5, 7 or 7, 4 or 7, 5)} \\ & = {4 \over 20} \\ & = {1 \over 5} \end{align*}

(iv) The numbers on the ball, 1, 2, 4, 5 and 7 are not divisible by 3 or by 9. Thus the product of any two numbers is not divisible by 9

\begin{align*} \text{P(Difference of two numbers is less than 7)} & = {20 \over 20} \\ & = 1 \end{align*}

(v)

\begin{align*} \text{P(Product of two numbers is divisible by 9)} & = {0 \over 20} \\ & = 0 \end{align*}

Possible outcomes in the form (first card, second card):

(i)

\begin{align*} \text{P(Scores on both cards are the same)} & = \text{P(1, 1 or 5, 5)} \\ & = {2 \over 16} \\ & = {1 \over 8} \end{align*}

(ii) A prime number only has two factors, 1 and itself

\begin{align*} \text{P(Scores on both cards are prime numbers)} & = \text{P(2, 3 or 2, 5 or 5, 3 or 5, 5)} \\ & = {4 \over 16} \\ & = {1 \over 4} \end{align*}

(iii)

\begin{align*} \text{P(Sum of scores is odd)} & = \text{P(1, 0 or 2, 1 or 2, 3 or 2, 5 or 4, 1 or 4, 3 or 4, 5 or 5, 0)} \\ & = {8 \over 16} \\ & = {1 \over 2} \end{align*}

(iv)

\begin{align*} \text{P(Sum of scores is divisible by 5)} & = \text{P(2, 3 or 4, 1 or 5, 0 or 5, 5)} \\ & = {4 \over 16} \\ & = {1 \over 4} \end{align*}

(v)

\begin{align*} \text{P(Sum of scores is 6 or less)} & = \text{P(1, 0 or 1, 1 or 1, 3 or 1, 5 or 2, 0 or 2, 1 or 2, 3 or 4, 0 or 4, 1 or 5, 0 or 5, 1)} \\ & = {11 \over 16} \end{align*}

(vi)

\begin{align*} \text{P(Product of scores is not 0)} & = 1 - \text{P(Product of scores is 0)} \\ & = 1 - \text{P(1, 0 or 2, 0 or 3, 0 or 4, 0)} \\ & = 1 - {4 \over 16} \\ & = {3 \over 4} \end{align*}

(vii)

\begin{align*} \text{P(Product of scores is greater than 11)} & = \text{P(4, 3 or 4, 5 or 5, 3 or 5, 5)} \\ & = {4 \over 16} \\ & = {1 \over 4} \end{align*}

Possible outcomes in the form (colour on spinner, head/tail from fair coin):

(i)

\begin{align*} \text{P(Red on spinner and tail on coin)} & = {1 \over 6} \end{align*}

(ii)

\begin{align*} \text{P(Blue or yellow on spinner and head on coin)} & = {2 \over 6} \\ & = {1 \over 3} \end{align*}

Possible outcomes in the form (card from 1st bag, card from 2nd bag):

(i)

\begin{align*} \text{P(Two numbers are odd)} & = \text{P(1, 1 or 1, 7 or 3, 1 or 3, 7 or 5, 1 or 5, 7)} \\ & = {6 \over 9} \\ & = {2 \over 3} \end{align*}

(ii)

\begin{align*} \text{P(Two numbers are prime numbers)} & = \text{P(3, 2 or 3, 7 or 5, 2 or 5, 7)} \\ & = {4 \over 9} \end{align*}

(iii)

\begin{align*} \text{P(Sum greater than 4)} & = \text{P(1, 7 or 3, 2 or 3, 7 or 5, 1 or 5, 2 or 5, 7)} \\ & = {6 \over 9} \\ & = {2 \over 3} \end{align*}

(iv)

\begin{align*} \text{P(Sum that is even)} & = \text{P(1, 1 or 1, 7 or 3, 1 or 3, 7 or 5, 1 or 5, 7)} \\ & = {6 \over 9} \\ & = {2 \over 3} \end{align*}

(v) A prime number only has two factors, 1 and itself

\begin{align*} \text{P(Product that is prime)} & = \text{P(1, 2 or 1, 7 or 3, 1 or 5, 1)} \\ & = {4 \over 9} \end{align*}

(vi)

\begin{align*} \text{P(Product that is greater than 20)} & = \text{P(3, 7 or 5, 7)} \\ & = {2 \over 9} \end{align*}

(vii)

\begin{align*} \text{P(Product that is divisible by 7)} & = \text{P(1, 7 or 3, 7 or 5, 7)} \\ & = {3 \over 9} \\ & = {1 \over 3} \end{align*}

(a)(i)

\begin{align*} \text{P(Face '4' down)} & = {1 \over 4} \end{align*}

(a)(ii) For this to occur, the odd number (either 1 or 3) must be facing down such that the sum of two even numbers and the remaining odd number gives an odd number

\begin{align*} \text{P(Sum of three upper faces is an odd number)} & = \text{P('1' down or '3' down)} \\ & = {2 \over 4} \\ & = {1 \over 2} \end{align*}

(b)

\begin{align*} \text{P(First component tested is defective)} & = {1 \over 7} \end{align*}

(a)(i)

(a)(ii)

\begin{align*} \text{P(Nora and Kate stay on different storeys)} & = {3 + 3 + 3 + 3 + 3 + 3 \over 30} \\ & = {3 \over 5} \end{align*}

(a)(iii)

\begin{align*} \text{P(Nora and Kate do not stay next to each other)} & = 1 - \text{P(Nora and Kate stay next to each other)} \\ & = 1 - {4 \over 15} \\ & = {11 \over 15} \end{align*}

(b)

\begin{align*} \text{P(Kate on second floor and next to Nora)} & = {1 + 2 + 1 \over 30} \\ & = {2 \over 15} \end{align*}