NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Conditional-probability simulations

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A craft designer in Brisbane is creating a pattern using two types of tiles: 51 blue tiles and 47 green tiles. The tiles are randomly arranged in a row.

The designer wants to simulate the probability that a green tile is immediately followed by a blue tile, given that a tile is not the last in the row. Using a simulation model with 1000 repeated trials, what is the exact probability (as a simplified fraction) that a green tile is followed by a blue tile in a random arrangement of all 98 tiles?

Assume each arrangement is equally likely and the simulation accounts for all possible sequences. Answer as a simplified fraction.

Concept illustrations

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Restrict the sample space before counting. All 12 dots are cyclists. Three filled dots also represent red clothing. Within the cyclist group, P(red given cycles) = 3/12 = 1/4.12 cyclists · 3 wearing red
All 12 dots are cyclists. Three filled dots also represent red clothing. Within the cyclist group, P(red given cycles) = 3/12 = 1/4.
A fair-coin simulation model. Choose 0 or 1 with equal probability: 0 means tails and 1 means heads. Repeat independent trials and count outcomes. No experiment results are asserted here.Choose 0 or 1Equal chance→0=T 1=HMap outcome→RepeatCount heads
Choose 0 or 1 with equal probability: 0 means tails and 1 means heads. Repeat independent trials and count outcomes. No experiment results are asserted here.

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Worked solution

  1. There are 98 tiles in total: 51 blue (B) and 47 green (G). The tiles are arranged randomly in a row, and each arrangement is equally likely.
  2. We are to find the probability that a green tile is immediately followed by a blue tile, given that the tile is not the last in the row. This means we are only considering positions 1 through 97 (since the 98th tile cannot be followed by another tile).
  3. The total number of adjacent tile pairs (i.e., positions (1,2), (2,3), ..., (97,98)) is 97. We are interested in how many of these pairs consist of a green tile followed by a blue tile (G then B).
  4. Since all arrangements are equally likely, we can compute the exact probability by counting the number of arrangements where a green tile is immediately followed by a blue tile, divided by the total number of arrangements. However, because the simulation is repeated 1000 times and the question asks for the exact probability, we must compute it analytically.
  5. The total number of ways to arrange 51 blue and 47 green tiles is the multinomial coefficient: 98! / (51! × 47!). This is large, but we do not need to compute it directly.
  6. We now compute the expected number of G-B adjacent pairs across all arrangements. For any given position i (from 1 to 97), the probability that tile i is green and tile i+1 is blue is: P(G at i) × P(B at i+1 | G at i).
  7. P(G at i) = 47 / 98, since 47 out of 98 tiles are green. Given that tile i is green, there are 97 tiles left, of which 51 are blue. So P(B at i+1 | G at i) = 51 / 97.
  8. Therefore, the probability that a specific pair (i, i+1) is G-B is (47/98) × (51/97).
  9. Since there are 97 such adjacent pairs, and each has the same probability of being G-B, the expected number of G-B pairs is 97 × (47/98) × (51/97). The 97 cancels out.
  10. Simplifying: (47/98) × 51 = (47 × 51) / 98. Compute 47 × 51 = 2397. So the expected number of G-B pairs is 2397 / 98.
  11. The probability that a randomly selected non-last tile (i.e., one of the 97) is followed by a blue tile, given it is green, is not directly the same as the expected number of G-B pairs. But since we are asked for the probability that a green tile is followed by a blue tile (conditioned on being green and not last), we must compute the conditional probability: P(G-B | G and not last).
  12. The total number of green tiles that are not in the last position is 47 minus the probability that a green tile is in the last position. But since all positions are symmetric, the probability that any green tile is in the last position is 47/98. So the expected number of green tiles not in the last position is 47 × (97/98). The expected number of G-B pairs is 2397 / 98. Therefore, the conditional probability is (2397 / 98) / (47 × 97 / 98) = 2397 / (47 × 97). Compute 47 × 97 = 4559. So 2397 / 4559. Simplify this fraction.
  13. Find GCD of 2397 and 4559. Use Euclidean algorithm: 4559 ÷ 2397 = 1 remainder 2162; 2397 ÷ 2162 = 1 remainder 235; 2162 ÷ 235 = 9 remainder 53; 235 ÷ 53 = 4 remainder 23; 53 ÷ 23 = 2 remainder 7; 23 ÷ 7 = 3 remainder 2; 7 ÷ 2 = 3 remainder 1; 2 ÷ 1 = 2 remainder 0. GCD is 1. So 2397/4559 is already simplified.

Reference answer: 2397/4559

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