8 out of every 10 persons who contract an infection recover. If 7 become infected, whats the probability tha 3 recover

The problem deals with binomial probability because we are calculating the probability of a specific number of successes (recovery) in a fixed number of trials (infected persons), where each trial has two possible outcomes (recover or not recover).

Let’s define the variables:
- \( n = 7 \) (the number of infected people),
- \( p = 0.8 \) (the probability of recovering, 8 out of 10),
- \( q = 1 - p = 0.2 \) (the probability of not recovering),
- \( k = 3 \) (the number of people who recover, as asked in the problem).

The binomial probability formula is:

\[
P(k \text{ recover}) = \binom{n}{k} \cdot p^k \cdot q^{n-k}
\]

Where:
- \( \binom{n}{k} \) is the binomial coefficient: \( \frac{n!}{k!(n-k)!} \),
- \( p^k \) is the probability that exactly \( k \) people recover,
- \( q^{n-k} \) is the probability that \( n-k \) people do not recover.

Let's calculate it step by step.

### Step 1: Binomial coefficient
\[
\binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
\]

### Step 2: Calculate the powers
\[
p^k = 0.8^3 = 0.512
\]
\[
q^{n-k} = 0.2^{4} = 0.0016
\]

### Step 3: Final calculation
\[
P(3 \text{ recover}) = 35 \cdot 0.512 \cdot 0.0016 = 0.028672
\]

So, the probability that exactly 3 out of the 7 people recover is approximately **0.0287** or **2.87%**.

Would you like a deeper explanation of this result?

Here are 5 related questions for further practice:
1. What is the probability that exactly 4 out of 7 people recover?
2. What is the probability that all 7 people recover?
3. What is the probability that none of the 7 people recover?
4. What is the expected number of people to recover out of 7 infected?
5. How does the probability change if the recovery rate drops to 70%?

**Tip:** Binomial probabilities are useful when dealing with situations where events are independent and have only two possible outcomes.

8 out of every 10 persons who contract an infection recover. If 7 become infected, what's the probability that 3 recover?

Calculate the probability that exactly 3 out of 7 infected people recover using the binomial probability formula. This problem covers key probability concepts, including binomial distribution, and is suitable for students in Grades 9-12. The recovery rate is 80%, and we explore step-by-step how to determine the probability of recovery.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation