Question

Ребятки помогите решить. " В партии готовой продукции из 15 лампочек 5 повышенного качества. наудачу берут 7 лампочек. какова вероятность того, что 3 лампочки в выборке будут повышенного качества?"

Answer 1

Problem Analysis

We are given a party with 15 light bulbs, out of which 5 are of higher quality. We randomly select 7 light bulbs from the party. We need to find the probability that exactly 3 of the selected light bulbs are of higher quality.

Solution

To solve this problem, we can use the concept of combinations and probability.

The total number of ways to select 7 light bulbs from a party of 15 is given by the combination formula:

C(n, r) = n! / (r! * (n-r)!)

Where: - n is the total number of light bulbs in the party (15 in this case) - r is the number of light bulbs we want to select (7 in this case) - ! denotes the factorial operation

Using this formula, we can calculate the total number of ways to select 7 light bulbs from the party.

Next, we need to calculate the number of ways to select exactly 3 light bulbs of higher quality from the 5 available. This can be calculated using the combination formula as well:

C(5, 3) = 5! / (3! * (5-3)!)

Finally, we can calculate the probability of selecting exactly 3 light bulbs of higher quality by dividing the number of ways to select 3 higher quality light bulbs by the total number of ways to select 7 light bulbs:

Probability = C(5, 3) / C(15, 7)

Let's calculate this probability.

Calculation

Using the combination formula, we can calculate the total number of ways to select 7 light bulbs from a party of 15:

C(15, 7) = 15! / (7! * (15-7)!) = 6435

Next, we calculate the number of ways to select exactly 3 light bulbs of higher quality from the 5 available:

C(5, 3) = 5! / (3! * (5-3)!) = 10

Finally, we calculate the probability:

Probability = C(5, 3) / C(15, 7) = 10 / 6435 ≈ 0.00155

Therefore, the probability that exactly 3 light bulbs in the selection of 7 will be of higher quality is approximately 0.00155.

Answer

The probability that exactly 3 light bulbs in the selection of 7 will be of higher quality is approximately 0.00155.

Note: The above calculation assumes that the selection of light bulbs is done without replacement.