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A bag contains n metal coins, n ≥ 3, that are made from either silver or bronze - HSC - SSCE Mathematics Extension 1 - Question 9 - 2024 - Paper 1
Question 9
A bag contains n metal coins, n ≥ 3, that are made from either silver or bronze. There are k silver coins in the bag and the rest are bronze. Two coins are to be d... show full transcript
Worked Solution & Example Answer:A bag contains n metal coins, n ≥ 3, that are made from either silver or bronze - HSC - SSCE Mathematics Extension 1 - Question 9 - 2024 - Paper 1
Step 1
Step 2
Calculate Probability for Silver Coins
Answer
The probability of drawing two silver coins can be computed as follows:
- The probability of drawing the first silver coin: ( P(S_1) = \frac{k}{n} )
- After drawing one silver coin, the probability of drawing another silver coin: ( P(S_2 | S_1) = \frac{k - 1}{n - 1} )
Thus, the total probability of drawing two silver coins is: [ P(S) = \frac{k}{n} \times \frac{k - 1}{n - 1} = \frac{k(k - 1)}{n(n - 1)} ]
Step 3
Calculate Probability for Bronze Coins
Answer
Similarly, the probability of drawing two bronze coins is:
- The probability of drawing the first bronze coin: ( P(B_1) = \frac{n - k}{n} )
- After drawing one bronze coin, the probability of drawing another bronze coin: ( P(B_2 | B_1) = \frac{n - k - 1}{n - 1} )
Thus, the total probability of drawing two bronze coins is: [ P(B) = \frac{n - k}{n} \times \frac{n - k - 1}{n - 1} = \frac{(n - k)(n - k - 1)}{n(n - 1)} ]
Step 4
Total Probability of Drawing Two Coins of Same Metal
Answer
Therefore, the total probability of drawing two coins of the same metal (either silver or bronze) is: [ P(S) + P(B) = \frac{k(k - 1)}{n(n - 1)} + \frac{(n - k)(n - k - 1)}{n(n - 1)} ]
To simplify: [ P(S + B) = \frac{k(k - 1) + (n - k)(n - k - 1)}{n(n - 1)} ]
This matches option A in the original question.