Assume that 9​% of people are​ left-handed. If 6 people are selected at​ random, find the probability of each outcome described below. Find the probability that there are some lefties among the 6 people.

We are given that 9% of people are left-handed, which means the probability of selecting a left-handed person is \( p = 0.09 \). The probability of selecting a right-handed person is \( 1 - p = 0.91 \). We are asked to find the probability that there are some left-handed people among 6 randomly selected individuals.

This is a binomial probability problem where:
- The number of trials \( n = 6 \),
- The probability of success (selecting a left-handed person) \( p = 0.09 \),
- The probability of failure (not selecting a left-handed person) \( 1 - p = 0.91 \).

### Step 1: Complementary Probability Approach
The easiest way to find the probability that there are some left-handed people is to use the complement rule. First, we calculate the probability that **none** of the 6 people are left-handed, then subtract this value from 1 to find the probability that at least one person is left-handed.

The probability that none of the 6 people are left-handed is:
\[
P(\text{none left-handed}) = (0.91)^6
\]

Now, calculate this:
\[
P(\text{none left-handed}) = 0.91^6 \approx 0.606
\]

### Step 2: Find the Probability of "Some Left-Handed"
The probability of having **some** left-handed people is the complement of the probability that none are left-handed:
\[
P(\text{some left-handed}) = 1 - P(\text{none left-handed})
\]
\[
P(\text{some left-handed}) = 1 - 0.606 = 0.394
\]

Thus, the probability that there are some left-handed people among the 6 selected is approximately **0.394**, or 39.4%.

Would you like more details or further questions?

Here are 5 related questions to explore:
1. What is the probability that exactly 2 of the 6 people are left-handed?
2. What is the probability that all 6 people are right-handed?
3. How does the binomial distribution change if the probability of being left-handed increases to 20%?
4. How many left-handed people would you expect to find among 10 randomly selected people?
5. What is the probability that fewer than 3 people are left-handed among 8 selected individuals?

**Tip:** The complement rule is often helpful when finding the probability of "at least one" event happening!

Assume that 9% of people are left-handed. If 6 people are selected at random, find the probability of each outcome described below. Find the probability that there are some lefties among the 6 people.

Learn how to calculate the probability of having some left-handed people among 6 randomly selected individuals, given that 9% of people are left-handed. This problem explores the binomial distribution and the complement rule, providing a detailed step-by-step solution. Suitable for students in Grades 9-12.

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