SSCE HSC Mathematics Extension 1 Bernoulli trials: Three squares are chosen at rand

Three squares are chosen at random from the 3 × 3 grid below, and a cross is placed in each chosen square - HSC - SSCE Mathematics Extension 1 - Question 10 - 2017 - Paper 1

Question 10

Three squares are chosen at random from the 3 × 3 grid below, and a cross is placed in each chosen square. What is the probability that all three crosses lie in the... show full transcript

Worked Solution & Example Answer:Three squares are chosen at random from the 3 × 3 grid below, and a cross is placed in each chosen square - HSC - SSCE Mathematics Extension 1 - Question 10 - 2017 - Paper 1

Step 1

Part a: Total Ways to Choose Squares

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Answer

To find the total number of ways to choose 3 squares from a 3 × 3 grid, we can use the combination formula:

$C(n, r) = \frac{n!}{r!(n-r)!}$

Here, ( n = 9 ) (total squares) and ( r = 3 ) (squares to choose). Therefore:

$C(9, 3) = \frac{9!}{3!6!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$

Step 2

Part b: Favorable Outcomes

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Answer

Next, we calculate the favorable outcomes where the three crosses lie in the same row, column, or diagonal:

Rows: There are 3 rows in the grid, and each row has exactly 3 squares.
Columns: Similarly, there are 3 columns in the grid, with 3 squares each.
Diagonals: There are 2 diagonals in the grid that can also contain 3 crosses.

Thus, the total favorable outcomes are:

$3 \text{ (rows)} + 3 \text{ (columns)} + 2 \text{ (diagonals)} = 8$

Step 3

Part c: Probability Calculation

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The probability is calculated by dividing the number of favorable outcomes by the total outcomes:

$P = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{8}{84} = \frac{2}{21}$

Three squares are chosen at random from the 3 × 3 grid below, and a cross is placed in each chosen square - HSC - SSCE Mathematics Extension 1 - Question 10 - 2017 - Paper 1

Worked Solution & Example Answer:Three squares are chosen at random from the 3 × 3 grid below, and a cross is placed in each chosen square - HSC - SSCE Mathematics Extension 1 - Question 10 - 2017 - Paper 1

Part a: Total Ways to Choose Squares

Part b: Favorable Outcomes

Part c: Probability Calculation

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