13 percent are left handed. 5 randomly selected. What is the probability of there being exactlcy 3 lefties in the group

This is a binomial probability problem, where we want to find the probability of exactly 3 left-handed people out of 5 randomly selected individuals.

The binomial probability formula is:

\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

Where:
- \(n = 5\) (the total number of trials, i.e., the number of people selected),
- \(k = 3\) (the number of successful outcomes, i.e., 3 left-handed people),
- \(p = 0.13\) (the probability of success on a single trial, i.e., being left-handed),
- \(\binom{n}{k}\) is the binomial coefficient, representing the number of ways to choose 3 lefties out of 5 people.

### Step-by-Step Calculation:

1. **Binomial coefficient** \(\binom{n}{k}\):
   \[
   \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10
   \]

2. **Calculate \(p^k\)**:
   \[
   p^k = 0.13^3 = 0.13 \times 0.13 \times 0.13 = 0.002197
   \]

3. **Calculate \((1 - p)^{n-k}\)**:
   \[
   (1 - p)^{n-k} = 0.87^2 = 0.87 \times 0.87 = 0.7569
   \]

4. **Multiply all parts together**:
   \[
   P(X = 3) = 10 \times 0.002197 \times 0.7569 = 0.01663
   \]

### Final Answer:
The probability of having exactly 3 left-handed people in the group is approximately **0.01663**, or **1.663%**.

Would you like more details or have any questions?

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Here are 5 related questions:
1. What is the probability of having at least 3 left-handed people in the group?
2. What is the probability of having no left-handed people in the group?
3. How does the probability change if 10 people are selected instead of 5?
4. How would you calculate the expected number of left-handed people in this group?
5. What is the probability of having exactly 2 left-handed people?

**Tip:** When dealing with binomial problems, always make sure you identify the number of trials, success probability, and how many successes you're looking for.

13 percent are left handed. 5 randomly selected. What is the probability of there being exactly 3 lefties in the group?

Calculate the probability of exactly 3 left-handed individuals out of 5 randomly selected people, using the binomial probability formula. This problem covers key probability concepts, including binomial distribution and the binomial coefficient. Suitable for students in Grades 9-12.

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