Calculating the Probability of Drawing at Least One Nestle Chocolate

Today, we will explore the fascinating world of probability using a simple problem involving a bag of chocolates. Let's consider the scenario where Ravi has a bag full of 20 Nestle and 5 Cadbury chocolates. If Ravi draws two chocolates from the bag, what is the probability that he would get at least one Nestle chocolate? We will utilize the complementary probability approach to solve this problem.

Step-by-Step Solution

The total number of chocolates in Ravi's bag is:

[ 20 text{ Nestle} 5 text{ Cadbury} 25 text{ chocolates} ]

The number of ways to choose 2 chocolates from 25 can be calculated using the combination formula:

[ binom{25}{2} frac{25 times 24}{2} 300 ]

To find the probability of not getting any Nestle chocolates (drawing 2 Cadbury chocolates), we need to calculate the number of ways to choose 2 Cadbury chocolates from the 5 available:

[ binom{5}{2} frac{5 times 4}{2} 10 ]

The probability of drawing 2 Cadbury chocolates is then:

[ P(text{no Nestle}) frac{text{Number of ways to choose 2 Cadbury}}{text{Total ways to choose 2 chocolates}} frac{10}{300} frac{1}{30} ]

Using the complementary probability approach, we can now find the probability of getting at least one Nestle chocolate:

[ P(text{at least one Nestle}) 1 - P(text{no Nestle}) 1 - frac{1}{30} frac{29}{30} ]

Alternative Method

Instead of using the complementary probability, we can calculate the probability of obtaining at least one Nestle chocolate directly:

[ P(text{at least one Nestle}) binom{20}{2} binom{20}{1} times binom{5}{1} binom{5}{1} times binom{20}{1} ]

Breaking it down:

[ binom{20}{2} frac{20 times 19}{2} 190 ] [ binom{20}{1} times binom{5}{1} 20 times 5 100 ] [ binom{5}{1} times binom{20}{1} 5 times 20 100 ]

Adding these together:

[ 190 100 100 290 ]

Dividing by the total number of ways to choose 2 chocolates:

[ frac{290}{300} frac{29}{30} ]

Conclusion

In conclusion, the probability that Ravi will draw at least one Nestle chocolate when he draws two chocolates from the bag is 29/30. This method showcases the power of both complementary probability and direct calculation in solving probability problems.

To summarize, the key concepts used in this calculation include:

Complementary probability Combination formula

These techniques are fundamental in combinatorics and probability theory and can be applied to a wide range of scenarios in everyday life and various fields of study.