Probability of Selecting Fresh Milk Packs from a Carton

In this article, we will explore the probability of selecting fresh milk packs from a carton containing a specific number of fresh and non-fresh milk packs. We will use combinatorial methods to solve the given problems step by step.

Problem Setup

A carton contains 12 milk packs, out of which 8 are fresh. A client selects 3 milk packs without replacement. We need to find:

The probability that the client will get exactly 2 fresh packs. The probability that the client will get at least 2 fresh packs.

Step-by-Step Solution

Let's break down the solution into detailed steps.

Total Milk Packs and Fresh Packs

Given:

Total milk packs 12 Fresh milk packs 8 Non-fresh milk packs 4 (12 - 8 4) Packs selected 3

Probability of Getting Exactly Two Fresh Packs

To find the probability of selecting exactly 2 fresh packs, we use the formula:

P ext{exactly 2 fresh} frac{binom{8}{2} times binom{4}{1}}{binom{12}{3}}

Calculate the number of ways to choose 2 fresh packs from 8:

binom{8}{2} frac{8!}{2! (8-2)!} frac{8 times 7}{2 times 1} 28

Calculate the number of ways to choose 1 non-fresh pack from 4:

binom{4}{1} frac{4!}{1! (4-1)!} 4

Total ways to choose 3 packs from 12:

binom{12}{3} frac{12!}{3! (12-3)!} frac{12 times 11 times 10}{3 times 2 times 1} 220

Calculate the total ways to choose 2 fresh and 1 non-fresh:

Total ways binom{8}{2} times binom{4}{1} 28 times 4 112

Calculate the probability:

P ext{exactly 2 fresh} frac{112}{220} frac{28}{55} approx 0.5091

Probability of Getting at Least Two Fresh Packs

To find the probability of getting at least 2 fresh packs, we sum the probabilities of getting exactly 2 fresh packs and exactly 3 fresh packs.

We already calculated the probability of getting exactly 2 fresh packs, which is 28/55.

Now, calculate the probability of getting exactly 3 fresh packs:

P ext{exactly 3 fresh} frac{binom{8}{3}}{binom{12}{3}}

Calculate the number of ways to choose 3 fresh packs from 8:

binom{8}{3} frac{8!}{3! (8-3)!} frac{8 times 7 times 6}{3 times 2 times 1} 56

Using the total ways to choose 3 packs 220:

P ext{exactly 3 fresh} frac{56}{220} frac{28}{110} frac{14}{55} approx 0.2545

Now sum these probabilities:

P ext{at least 2 fresh} P ext{exactly 2 fresh} P ext{exactly 3 fresh} frac{28}{55} frac{14}{55} frac{42}{55} approx 0.7636

Final Answers

The probability that the client will get exactly 2 fresh packs is:

frac{28}{55} approx 0.5091

The probability that the client will get at least 2 fresh packs is:

frac{42}{55} approx 0.7636