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

To find the probability that both bulbs chosen can be used to light the room, we need to determine the number of favorable outcomes and the total number of outcomes. 

1. Calculate the total number of ways to choose 2 bulbs out of 9 bulbs:
   \(\dbinom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36\).
2. Determine the number of usable (non-fused) bulbs. Out of 9 bulbs, 2 are fused, so the remaining usable bulbs are \(9 - 2 = 7\).
3. Calculate the number of ways to choose 2 usable bulbs from these 7 bulbs:
   \(\dbinom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21\).
4. Therefore, the probability of choosing 2 bulbs that can be used to light the room is given by the ratio of favorable outcomes to total outcomes:
   \(\frac{21}{36}\).
5. Simplify the fraction:
   \(\frac{21}{36} = \frac{7}{12}\).

Hence, the probability that both bulbs chosen can be used to light the room is \(\frac{7}{12}\), which matches the provided correct answer.

Answer 2

Step 1: Work out how many bulbs actually work.
There are 9 bulbs in total and 2 are fused, so \(9 - 2 = 7\) bulbs work properly.

Step 2: Count the total ways to pick any 2 bulbs from the bag.
Total ways = \(\binom{9}{2} = \dfrac{9 \times 8}{2} = 36\).

Step 3: Count the favourable ways, where both bulbs picked are good.
Favourable ways = \(\binom{7}{2} = \dfrac{7 \times 6}{2} = 21\), since both bulbs must come from the 7 working bulbs.

Step 4: Divide favourable ways by total ways.
Probability = \(\dfrac{21}{36} = \dfrac{7}{12}\).

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