### More about probability - (15.1-15.3)

#### Recognise the notation of set language including union, intersection and complement

• There are 4 cards marked as ‘H’, ‘O’, ‘U’ and ‘R’ each. 2 cards are drawn from the 4 cards without replacement.

(a) Find the sample space S and the corresponding size n(S).
(b) Express each of the following events as a set and find the corresponding size.
(i) A, where A is the event that the letter on the first card is a vowel
(ii) B, where B is the event that the letter on the second card is a vowel
(iii) C, where C is the event that the letters on both cards are vowels
(iv) $$A \cap B$$
(v) $$B \cap C$$
(vi) $$\bar B \cap C$$
• (a) S ={HO, HU, HR, OH, OU, OR, UH, UO,UR, RH, RO, RU}
$$n(S) = \underline{\underline {12}}$$

(b) (i) A= {OH, OU, OR, UH, UO, UR}
$$n(A) = \underline{\underline 6}$$

(ii) B= {HO, HU, OU, UO, RO, RU}
$$n(B) = \underline{\underline 6}$$

(iii) C ={OU, UO}
$$n(C) = \underline{\underline 2}$$

(iv)$$A \cap B$$ = {OU, UO}
$$n(A \cap B) = \underline{\underline 2}$$

(v) $$B \cap C$$ ={OU, UO}
$$n(B \cap C) = \underline{\underline 2}$$

(vi)$$\because \bar B = \{ {\rm{H R , OH , OR , U H , U R , R H}}\}$$
$$\therefore \bar B \cap C = \underline{\underline \varphi }$$
$$n(\bar B \cap C) = \underline{\underline 0}$$

#### Understand the addition law of probability and the concepts of mutually exclusive events and complementary events

• a) In a game, each participant needs to draw a ball with a number on it from a
box. It is given that the probabilities of the number on the ball drawn is smaller
than 10, greater than 50, and between 15 and 30 are $$\frac{1}{{24}}$$, $$\frac{5}{{12}}$$ and $$\frac{7}{{18}}$$ respectively. If a
ball is drawn at random, find the probability of drawing a ball with a number that
is smaller than 10 or greater than 50.

b)Two fair dice are thrown once. Find the probability that the product of the numbers
obtained is smaller than 25.
• a) P(Smaller than 10 or greater than 50)
= P(Smaller than 10) + P(Greater than 50)
$$\begin{array}{l} = \frac{1}{{24}} + \frac{5}{{12}}\\ = {\underline{\underline {\frac{{11}}{{24}}}} ^{}}\end{array}$$

b) P(Product is smaller than 25)
= 1 - P(Product is greater than or equal 25)
= 1 - P({(5, 5), (5, 6), (6, 5), (6, 6)})
$$\begin{array}{l} = 1 - \frac{4}{{36}}\\ = \underline{\underline {\frac{8}{9}}} \end{array}$$

#### Understand the multiplication law of probability and the concept of independent events

• A fair dice is tossed once and a letter is chosen from the word
GOGGLE at random. Find the probability that the number obtained
is smaller than 3 and the letter obtained is G.
• Let A be the event of obtaining a number smaller than 3,
B be the event of obtaining a letter G.
$$\begin{array}{c}P(A \cap B) = P(A) \times P(B)\\ = \frac{2}{6} \times \frac{3}{6}\\ = \underline{\underline {\frac{1}{6}}} \end{array}$$