Problem 19 - Algebra and Trigonometry

Question

Blood Type About 45\(\%\) of the population of the United States and Canada have Type \(\mathrm{O}\) blood. (a) If a random sample of ten people is selected, what is the probability that exactly five have Type O blood? (b) What is the probability that at least three of the random sample of ten have Type O blood?

Step-by-Step Solution

Answer

(a) 0.2001 (b) 0.8695

Step 1: Understand the Binomial Distribution

In this problem, we are dealing with a Binomial Distribution since we have a fixed number of trials (10 people), each trial has two possible outcomes (Type O blood or not), and the probability of success (having Type O blood) is constant (0.45).

Step 2: Define Variables for Part (a)

Let \( n = 10 \) be the number of people, \( k = 5 \) be the number of successes, and \( p = 0.45 \) be the probability of success (having Type O blood).

Step 3: Calculate Binomial Probability for Part (a)

The probability of exactly \( k \) (5) successes in \( n \) (10) trials is given by the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \].So, calculate: \[ P(X = 5) = \binom{10}{5} (0.45)^5 (0.55)^5 \].

Step 4: Compute the Combination and Probability for Part (a)

Calculate the combination \( \binom{10}{5} = 252 \), then substitute into the formula:\[ P(X = 5) = 252 \times (0.45)^5 \times (0.55)^5 \approx 0.2001 \].

Step 5: Define Variables for Part (b)

For part (b), we need to find the probability of at least 3 successes: \( X \geq 3 \). It is often easier to calculate \( P(X < 3) \) and subtract from 1.

Step 6: Calculate Probability for Fewer Than 3 Successes

Compute \( P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) \). Use the formula for each: \[ P(X = 0) = \binom{10}{0} (0.45)^0 (0.55)^{10} \approx 0.0035 \],\[ P(X = 1) = \binom{10}{1} (0.45)^1 (0.55)^9 \approx 0.0285 \],\[ P(X = 2) = \binom{10}{2} (0.45)^2 (0.55)^8 \approx 0.0985 \],then, \( P(X < 3) = 0.0035 + 0.0285 + 0.0985 \approx 0.1305 \).

Step 7: Calculate Probability for At Least 3 Successes

Use \( P(X \geq 3) = 1 - P(X < 3) \), so: \[ P(X \geq 3) = 1 - 0.1305 \approx 0.8695 \].

Key Concepts

Probability

Probability in statistics is the concept that measures how likely an event is to occur. It ranges from 0 to 1, with 0 meaning no chance and 1 indicating certainty. In the context of our exercise, understanding probability helps us calculate the likelihood of a particular outcome, such as how many people in a group of ten might have Type O blood.

For Question (a), we aim to find the probability that exactly five people have Type O blood. This requires us to use the binomial probability formula, which applies to situations with fixed numbers of trials and exactly two possible outcomes for each trial. The probability was calculated by determining the probability of success (getting a person with Type O blood) and applying it to the set number of trials (people sampled).

For Question (b), we calculated the probability of having at least a certain number (three or more) of successes in a sample of ten. This involved finding the complementary probability of having fewer successes and subtracting from one.

Combination

Combination mathematics is a crucial concept when dealing with probabilities, particularly in binomial distributions. It allows us to determine the number of different ways a particular event can occur. This is known as choosing, or selecting a specific number of successes from a set number of trials, symbolized as \( \binom{n}{k} \), which reads as 'n choose k'.

In our exercise, this concept is used to calculate the number of ways five people can be Type O blood out of a total of ten people. The formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) where '!' denotes a factorial, helps us determine this. Here, \( \binom{10}{5} \) equals 252, meaning there are 252 ways five Type O blood individuals could be found in a sample of ten. Combinations play a vital role in the calculation of binomial probabilities by determining how choices are made in scenarios with multiple possibilities.

Population Statistics

Population statistics involve the study and analysis of the characteristics of populations, which allows us to infer patterns and distributions.

In the context of our example, knowing that 45\( \% \) of the population has Type O blood is a critical statistic. This figure represents the probability of any single person having Type O blood and is used as the 'success' probability in our binomial model. With a representative percentage, statisticians can predict outcomes for samples and understand the variability within larger groups.

The population statistic here is crucial, as it directly informs the success probability \( p = 0.45 \) that is used in the computation of probabilities for individuals in smaller samples, like the random selection of ten people discussed in the exercise.

Blood Type Distribution

Blood type distribution refers to how common each blood type is across a specific population. This information is vital for various fields, including medicine and genetics.

In this exercise, recognizing that 45\( \% \) of individuals have Type O blood in the U.S. and Canada provides a foundational datum for understanding broader and more targeted population trends. Blood type distribution helps us in calculating the likelihood of encountering a particular blood type when sampling from a population.

This sort of statistical data aids in resource allocation, such as planning blood bank supplies to meet potential demand. In our problem, the distribution is used to understand and predict outcomes in smaller samples, which aligns directly with the principles of binomial distributions.