Question

The odds in favour of A winning a game of table tennis against B are 1:2. If 3 games are to be played, then the probability of A winning at least two games out of the three is.....

• A. \(\frac{1}{27}\)
• B. \(\frac{2}{27}\)
• C. \(\frac{5}{27}\)
• D. \(\frac{7}{27}\)

Hint:

Convert the odds to a probability of winning, then use the binomial formula for 2 or 3 wins.

Answer 1

The Correct Option is D

Step 1: Use the complement.
$P(X \geq 2) = 1 - P(X = 0) - P(X = 1)$ with $p = 1/3$.

Step 2: Compute.
$P(X = 0) = \left(\dfrac{2}{3}\right)^3 = \dfrac{8}{27}$ and $P(X = 1) = 3\cdot\dfrac{1}{3}\cdot\dfrac{4}{9} = \dfrac{12}{27}$.

Step 3: Subtract.
\[ 1 - \frac{8}{27} - \frac{12}{27} = \frac{7}{27} \]

Final Answer:
Option (D). \[ \boxed{\frac{7}{27}} \]