Jacob, Amelie and Reuben each roll a fair six-sided dice - OCR - GCSE Maths - Question 3 - 2019 - Paper 1

To find the probability that all three players roll a number less than 3, we first identify the outcomes for a single die roll. A fair six-sided die has the numbers 1 through 6. The numbers less than 3 are 1 and 2.

1. Calculate the probability for one die: The favorable outcomes are 1 and 2. Therefore, the probability of rolling a number less than 3 for a single die is:

   $P(X < 3) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2}{6} = \frac{1}{3}$

2. Calculate the probability for three dice: Since the dice rolls are independent events, we can find the combined probability by multiplying the individual probabilities:

   $P(X_1 < 3) \cdot P(X_2 < 3) \cdot P(X_3 < 3) = \left(\frac{1}{3}\right) \cdot \left(\frac{1}{3}\right) \cdot \left(\frac{1}{3}\right) = \frac{1}{27}$

Therefore, the probability that all three roll a number less than 3 is ( \frac{1}{27} ).