12 light bulbs are tested to see if they last as long as claimed. 3 of them fail. 2 light bulbs are selected at random without replacement. find the probablility that at least one of the 2 selected failed the test.
45%
54.5%
45.5%
To find the probability that at least one of the two selected light bulbs failed the test, we can calculate the probability of the complementary event, which is the probability that both selected light bulbs passed the test.
Let's start by calculating the probability that the first selected light bulb passed the test. Since 3 out of the 12 light bulbs failed, the probability of selecting a light bulb that passed is 9/12.
Next, we need to calculate the probability that the second selected light bulb passed the test, given that the first selected light bulb passed. Since we're selecting without replacement, there are now 11 light bulbs left, with 2 of them being failed bulbs. Therefore, the probability of selecting a second light bulb that passed is 9/11.
To calculate the probability that both selected light bulbs passed, we multiply the probabilities of each event (since they are independent):
P(both bulbs passed) = (9/12) * (9/11) = 3/4 * 9/11 = 27/44
Finally, to find the probability that at least one of the two selected light bulbs failed, we subtract the probability of the complementary event from 1:
P(at least one bulb failed) = 1 - P(both bulbs passed) = 1 - 27/44
Therefore, the probability that at least one of the two selected light bulbs failed the test is 17/44, or approximately 0.39.
To find the probability that at least one of the two selected light bulbs failed the test, we can use the concept of complementary probability.
First, let's find the probability of selecting two light bulbs that passed the test.
There are 12 light bulbs in total, and 3 of them failed. So, the number of light bulbs that passed the test is 12 - 3 = 9.
When selecting the first light bulb, the probability of selecting a light bulb that passed the test is 9/12.
After the first light bulb is selected, there are now 11 light bulbs remaining, with 8 of them passing. So, the probability of selecting a second light bulb that passed the test is 8/11.
To find the probability of both events happening (selecting two light bulbs that passed), we multiply the probabilities: (9/12) * (8/11) = 72/132 = 9/16.
Now, we can find the probability of at least one of the two selected light bulbs failing the test:
Since this is a complementary event, it is equal to 1 - probability of both bulbs passing.
So, the probability of at least one of the two selected light bulbs failing the test is 1 - 9/16 = 7/16.
Therefore, the probability that at least one of the two selected light bulbs failed the test is 7/16.