More about probability - (15.1-15.3)

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

  • Theory
  • Examples
    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) \(\bar B \cap C\)
  • 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
  • Examples
    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.
  • Solutions
    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

  • Theory
  • Examples
    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.
  • Solutions

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