9.5.1 Probability of Combined Events, SPM Paper (Long Questions)

Question 1:

All the cards written with the letters from the word ‘INTERNATIONAL’ are put into a box.

Two cards are drawn at random from the box, one after another, without replacement. Calculate the probability that

(a) the first card drawn has a letter N and the second card drawn has a letter I.

(b) the two cards drawn have the same letter.

Solution:
(a)
There are three cards with the letter ‘N’ and two card with letter ‘I’.
P (the first card drawn has a letter N and the second card drawn has a letter I)

\begin{array}{l} = \frac{3}{13} \times \frac{2}{(13 - 1)} \\ = \frac{3}{13} \times \frac{2}{12} = \frac{1}{26} \end{array}

(b)
P (the two cards drawn have the same letter)
= P (II or NN or TT or AA)
= P (II) + P (NN) + P (TT) + P (AA)

\begin{array}{l} = (\frac{2}{13} \times \frac{1}{12}) + (\frac{3}{13} \times \frac{2}{12}) + (\frac{2}{13} \times \frac{1}{12}) + (\frac{2}{13} \times \frac{1}{12}) \\ ↑ \\ \begin{array}{l} There are 3 letter ‘ N ‘ \\ out of 13 letters \end{array} \\ = \frac{2}{156} + \frac{6}{156} + \frac{2}{156} + \frac{2}{156} \\ = \frac{12}{156} \\ = \frac{1}{13} \end{array}

Question 2:
During National Day celebration, a group of 8 boys and 5 girls from a school are taking part in a singing competition. Each day, two pupils are chosen at random to perform special skill.

(a) Calculate the probability that both pupils chosen to perform special skill are boys.

(b) Two boys were chosen to perform special skill on the first day. They are exempted from performing special skill on the second day.
Calculate the probability that both pupils chosen to perform special skill on the second day are of the same gender.

Solution:
(a)

\begin{array}{l} P (both pupils are boys) \\ = P (B B) \\ = \frac{8}{13} \times \frac{7}{12} \\ = \frac{7}{24} \end{array}

(b)

\begin{array}{l} P (both pupils are of the same gender) \\ = P (B B) + P (G G) \\ = (\frac{6}{11} \times \frac{5}{10}) + (\frac{5}{11} \times \frac{4}{10}) \leftarrow \begin{array}{l} 11 pupils left in the group to \\ perform special skill on the second \\ day after two boys were exempted . \end{array} \\ = \frac{3}{11} + \frac{2}{11} \\ = \frac{5}{11} \end{array}

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