Question

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

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

Answer 1

The Correct Option is B

To solve the problem of determining the probability that both selected bulbs can be used to lighten the room, we should follow these steps:

1. Identify the total number of bulbs and the number of usable bulbs. We have 9 bulbs in total, and 2 are fused. Therefore, the number of usable bulbs is \(9 - 2 = 7\).
2. Calculate the total number of ways to select 2 bulbs out of 9. This can be done using the combination formula \(\binom{n}{r}\), where \(n\) is the total number of items, and \(r\) is the number of items to choose: \[\binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36.\]
3. Next, calculate the number of ways to select 2 usable bulbs out of the 7 that are not fused: \[\binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21.\]
4. The probability that both bulbs selected are usable is the ratio of the number of favorable outcomes to the total number of outcomes. Therefore, the probability \(P\) is: \[P = \frac{\binom{7}{2}}{\binom{9}{2}} = \frac{21}{36} = \frac{7}{12}.\]

Hence, the probability that both selected bulbs can be used is \(\frac{7}{12}\).