Calculating the Final Probability

Oct 13, 2025 by vinooalluri.com 95 views

**In the United States, $39\%$ of the Population has Type A Blood: A Binomial Probability Analysis**

In this article, we will explore the probability of a specific scenario involving type A blood in the United States population. We will use the formula for binomial probabilities to calculate the probability of exactly 8 people out of 10 randomly chosen individuals having type A blood.

Before we dive into the calculation, let's understand the concept of binomial probabilities. The binomial probability formula is used to calculate the probability of achieving 'k' successes in 'n' trials, where the probability of success in each trial is 'p'. The formula is given by:

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

where $\binom{n}{k}$ is the number of combinations of 'n' items taken 'k' at a time, also known as the binomial coefficient.

Now, let's apply the formula to our scenario. We have 10 people randomly chosen from the U.S. population, and we want to find the probability that exactly 8 of them have type A blood. We are given that $39\%$ of the population has type A blood, which means the probability of a person having type A blood is $p=0.39$ .

Using the binomial probability formula, we can calculate the probability as follows:

P(8)=\binom{10}{8} (0.39)^8(1-0.39)^{10-8}

Step-by-Step Calculation

To calculate the probability, we need to follow these steps:

Calculate the binomial coefficient $\binom{10}{8}$ .
Raise $0.39$ to the power of 8.
Raise $(1-0.39)$ to the power of 2.
Multiply the results from steps 1, 2, and 3.

Q: What is the probability of a person having type A blood in the United States? A: The probability of a person having type A blood in the United States is $39\%$ , which is equivalent to $0.39$ .

Q: How many people are randomly chosen from the U.S. population? A: 10 people are randomly chosen from the U.S. population.

Q: What is the probability of exactly 8 people out of 10 having type A blood? A: We will use the binomial probability formula to calculate this probability.

Q: What is the binomial coefficient $\binom{10}{8}$ ? A: The binomial coefficient $\binom{10}{8}$ is equal to 45.

Q: What is the value of $(1-0.39)$ ? A: The value of $(1-0.39)$ is equal to 0.61.

Q: What is the final probability of exactly 8 people out of 10 having type A blood? A: The final probability is equal to $P(8)=45 \times (0.39)^8 \times (0.61)^2$ .

Calculating the Final Probability

Now, let's calculate the final probability:

P(8)=45 \times (0.39)^8 \times (0.61)^2

P(8)=45 \times 0.0000513 \times 0.371601

P(8)=0.0081

In this article, we used the binomial probability formula to calculate the probability of exactly 8 people out of 10 randomly chosen individuals having type A blood in the United States. We found that the probability is approximately $0.0081$ . This means that the probability of exactly 8 people out of 10 having type A blood is very low, which is expected given the low prevalence of type A blood in the population.

Q: What is the binomial probability formula? A: The binomial probability formula is used to calculate the probability of achieving 'k' successes in 'n' trials, where the probability of success in each trial is 'p'. The formula is given by:

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

Q: What is the binomial coefficient $\binom{n}{k}$ ? A: The binomial coefficient $\binom{n}{k}$ is the number of combinations of 'n' items taken 'k' at a time.

Q: How many people are randomly chosen from the U.S. population? A: 10 people are randomly chosen from the U.S. population.

Q: What is the probability of exactly 8 people out of 10 having type A blood? A: We used the binomial probability formula to calculate this probability, and found that it is approximately $0.0081$ .