Probability Calculation for Same Colored Balls

The problem asks for the probability of picking 3 balls of the same colour from a box containing 12 balls.

• Total balls = 12
• Number of colours = 4 (Red, Green, Blue, Yellow)
• Number of balls of each colour = 3
• Number of balls picked = 3 (without replacement)

Total Possible Outcomes

The total number of ways to pick any 3 balls from the 12 available balls is calculated using combinations:

Total Combinations = $ \binom{12}{3} $

$ \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 2 \times 11 \times 10 = 220 $

So, there are 220 possible ways to pick 3 balls from the box.

Favorable Outcomes (All Same Colour)

For all 3 balls to be the same colour, we must pick 3 balls of a single colour. There are 4 colours to choose from.

For any specific colour (e.g., Red), there are 3 balls. The number of ways to pick 3 balls of that specific colour is:

Ways for one colour = $ \binom{3}{3} $

$ \binom{3}{3} = \frac{3!}{3!(3-3)!} = \frac{3!}{3!0!} = 1 $

Since there are 4 colours, the total number of ways to pick 3 balls of the same colour is:

Favourable Combinations = (Number of colours) $ \times $ (Ways to pick 3 of one colour)

Favourable Combinations = $ 4 \times \binom{3}{3} = 4 \times 1 = 4 $

Calculating the Probability

The probability is the ratio of favourable outcomes to the total possible outcomes:

Probability (All 3 same colour) = $ \frac{\text{Favourable Combinations}}{\text{Total Combinations}} $

$ P(\text{All 3 same colour}) = \frac{4}{220} $

Simplifying the fraction:

$ P(\text{All 3 same colour}) = \frac{1}{55} $