a box of light bulbs contains 95 good bulbs and 5 bad ones. if 3 bulbs are selected at random from the box, what is the probability that 2 are good and 1 is bad
To solve this problem, we can use the concept of combinations.
The total number of bulbs in the box is 95 + 5 = 100.
We need to select 3 bulbs from this box.
The number of combinations to select 3 bulbs at random from a set of 100 is given by:
C(100, 3) = 100! / (3! * (100 - 3)!)
= 100! / (3! * 97!)
= (100 * 99 * 98) / (3 * 2 * 1)
= 161,700
Now, let's find the number of combinations that have 2 good bulbs and 1 bad bulb.
The number of combinations to select 2 good bulbs from 95 is given by:
C(95, 2) = 95! / (2! * (95 - 2)!)
= 95! / (2! * 93!)
= (95 * 94) / (2 * 1)
= 4,470
The number of combinations to select 1 bad bulb from 5 is given by:
C(5, 1) = 5! / (1! * (5 - 1)!)
= 5! / (1! * 4!)
= 5 / 1
= 5
To find the total number of combinations with 2 good bulbs and 1 bad bulb, we multiply these two values:
Total combinations = C(95, 2) * C(5, 1)
= 4,470 * 5
= 22,350
Finally, to find the probability, we divide the total number of combinations with 2 good bulbs and 1 bad bulb by the total number of combinations of selecting 3 bulbs:
P(2 good, 1 bad) = Total combinations / C(100, 3)
= 22,350 / 161,700
≈ 0.138
To find the probability of selecting 2 good bulbs and 1 bad bulb, we need to first calculate the total number of possible outcomes and the number of favorable outcomes.
1. Total number of possible outcomes:
We can select 3 bulbs from a box that contains a total of 100 bulbs (95 good bulbs + 5 bad bulbs). Therefore, the total number of possible outcomes is given by the formula for combinations:
C(n, k) = n! / (k! * (n - k)!)
Here, n represents the total number of objects (100 bulbs) and k represents the number of objects being chosen (3 bulbs).
C(100, 3) = 100! / (3! * (100 - 3)!) = 100! / (3! * 97!) = (100 * 99 * 98) / (3 * 2 * 1) = 161,700
So, there are 161,700 possible outcomes when selecting 3 bulbs from the box.
2. Number of favorable outcomes:
We want to select 2 good bulbs from the 95 good bulbs and 1 bad bulb from the 5 bad bulbs. The number of favorable outcomes is given by the product of the number of ways to choose the bulbs:
C(95, 2) * C(5, 1) = (95! / (2! * (95 - 2)!) * 5! / (1! * (5 - 1)!) = (95 * 94 / 2) * 5 = 4,485
So, there are 4,485 favorable outcomes when selecting 2 good bulbs and 1 bad bulb.
3. Probability:
The probability is given by the ratio of the number of favorable outcomes to the total number of possible outcomes:
P = favorable outcomes / total outcomes = 4,485 / 161,700 ≈ 0.0277
Therefore, the probability that 2 bulbs are good and 1 bulb is bad is approximately 0.0277, or 2.77%.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
95/100 * 95/100 * 5/100 = ?