More about probability  (15.115.3)
Recognise the notation of set language including union, intersection and complement

Theory

ExamplesThere 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) \(\bar B \cap C\)

Solutions(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

Theory

Examplesa) 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. 
Solutionsa) 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

Theory

ExamplesA 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. 
SolutionsLet A be the event of obtaining a number smaller than 3,
B be the event of obtaining a letter G.
\(\begin{array}{c}P(A \cap B) = P(A) \times P(B)\\ = \frac{2}{6} \times \frac{3}{6}\\ = \underline{\underline {\frac{1}{6}}} \end{array}\)