Probability of Selecting at Least One Black Ball from a Basket

Suppose you have a basket containing 6 white balls, 4 black balls, 2 pink balls, and 3 green balls. We want to find the probability that at least one ball selected at random is black when four balls are picked.

Introduction to Probability and Combinatorics

In probability theory, we often use combinatorics to calculate the number of ways to choose items from a set. Combinatorics is the branch of mathematics concerned with counting, arranging, and listing objects of a finite set.

Methodology

To find the probability that at least one ball is black, we use complementary probability. This involves calculating the probability of the opposite event (picking no black balls) and then subtracting it from 1. This approach simplifies the calculation, especially when dealing with complex events.

Step 1: Total Number of Balls

First, let's calculate the total number of balls in the basket:

White balls: 6 Black balls: 4 Pink balls: 2 Green balls: 3

Total balls 6 4 2 3 15

Step 2: Total Ways to Choose 4 Balls

Next, we calculate the total number of ways to choose 4 balls from 15 balls using combinations (denoted as ( binom{n}{r} )):

[ binom{15}{4} frac{15!}{4!(15-4)!} frac{15 times 14 times 13 times 12}{4 times 3 times 2 times 1} 1365 ]

Step 3: Choosing 4 Balls with No Black Balls

The non-black balls consist of:

White balls: 6 Pink balls: 2 Green balls: 3

Total non-black balls 6 2 3 11

We calculate the number of ways to choose 4 non-black balls from these 11:

[ binom{11}{4} frac{11!}{4!(11-4)!} frac{11 times 10 times 9 times 8}{4 times 3 times 2 times 1} 330 ]

Step 4: Calculate the Probability of No Black Balls

The probability of choosing no black balls is:

[ P(text{no black balls}) frac{330}{1365} ]

Step 5: Calculate the Probability of At Least One Black Ball

The probability of getting at least one black ball is:

[ P(text{at least one black ball}) 1 - P(text{no black balls}) 1 - frac{330}{1365} ]

Calculate the final probability:

[ P(text{at least one black ball}) 1 - frac{330}{1365} frac{1365 - 330}{1365} frac{1035}{1365} ]

Step 6: Simplify the Fraction

Simplify the fraction:

[ frac{1035}{1365} frac{1035 div 105}{1365 div 105} frac{39}{51} frac{13}{17} ]

Final Answer

Therefore, the probability that at least one ball is black when picking 4 balls is:

[ boxed{frac{13}{17}} ]

Bonus Calculation

As an additional exercise, let's calculate the probability of picking a black ball in a simpler scenario. Given a sample space (S) with 461 total items and 11 black items:

Let (E) be the event of getting a black ball. Then:

[ nS 461 ] [ nE 11 ] [ P(E) frac{nE}{nS} frac{11}{461} approx 0.024 ]

This shows that in a smaller, simpler case, the probability of picking a black ball is approximately 0.024.