Question

A person has a bag which contains 9 bulbs, out of which 2 are fused and cannot be used to light the room. Two bulbs are selected at random. What is the probability that both bulbs chosen can be used to light the room?

• A. \(\dfrac{5}{12}\)
• B. \(\dfrac{7}{12}\)
• C. \(\dfrac{9}{12}\)
• D. \(\dfrac{10}{12}\)

Hint:

Use combinations: ways to choose 2 good bulbs over ways to choose any 2 bulbs.

Answer 1

The Correct Option is B

Step 1: Think of picking the two bulbs one after another instead of together.
There are 9 bulbs, 7 good and 2 fused. Pick the first bulb, then the second, without putting the first one back.

Step 2: Find the chance the first bulb picked is good.
$P(\text{first is good}) = \dfrac{7}{9}$.

Step 3: Find the chance the second bulb is also good, given the first one already was.
After removing one good bulb, 6 good bulbs remain out of 8 total, so $P(\text{second good} \mid \text{first good}) = \dfrac{6}{8} = \dfrac{3}{4}$.

Step 4: Multiply the two chances together.
$P(\text{both good}) = \dfrac{7}{9} \times \dfrac{3}{4} = \dfrac{21}{36} = \dfrac{7}{12}$, the same result as the combinations method.

Final Answer:
The probability both bulbs picked work is $\dfrac{7}{12}$. \[ \boxed{\dfrac{7}{12}} \]