Question

In the United States, \(44 \%\) of the population has type A blood. Consider taking a sample of size \(4 .\) Let \(Y\) denote the number of persons in the sample with type \(A\) blood. Find (a) \(\operatorname{Pr}[Y=0\\}\). (b) \(\operatorname{Pr}\\{Y=1\\}\). (c) \(\operatorname{Pr}\\{Y=2\\}\). (d) \(\operatorname{Pr}\\{0 \leq Y \leq 2\\}\). (e) \(\operatorname{Pr}\\{0

Short answer

Pr(Y=0) is 0.087, Pr(Y=1) is 0.335, Pr(Y=2) is 0.367, Pr(0≤Y≤2) is 0.789, Pr(0<Y≤2) is 0.702.

Step-by-step solution

Identify the Distribution

Since we are dealing with a fixed number of independent trials, each having two possible outcomes (type A blood or not), and the probability of success (type A blood) is the same for each trial, we are looking at a binomial distribution. The number of trials is 4, and the probability of success on a single trial is 0.44.

Calculate Pr(Y=0)

Using the binomial probability formula: \(Pr(Y=k) = C(n, k) \times p^k \times (1-p)^{(n-k)}\), where \(n\) is the number of trials, \(k\) is the number of successes, \(p\) is the probability of success, and \(C(n, k)\) is the number of combinations of \(n\) items taken \(k\) at a time. For \(Y=0\): \(Pr(Y=0) = C(4, 0) \times 0.44^0 \times (1-0.44)^{(4-0)}\).

Calculate Pr(Y=1)

Using the binomial probability formula for \(Y=1\): \(Pr(Y=1) = C(4, 1) \times 0.44^1 \times (1-0.44)^{(4-1)}\).

Calculate Pr(Y=2)

Using the binomial probability formula for \(Y=2\): \(Pr(Y=2) = C(4, 2) \times 0.44^2 \times (1-0.44)^{(4-2)}\).

Calculate Pr(0≤Y≤2)

This is the cumulative probability of \(Y\) taking values 0, 1, or 2. We find this by adding up the individual probabilities: \(Pr(0≤Y≤2) = Pr(Y=0) + Pr(Y=1) + Pr(Y=2)\).

Calculate Pr(0

For this conditional probability, exclude the chance of \(Y=0\): \(Pr(0<Y≤2) = Pr(Y=1) + Pr(Y=2)\).

Key concepts

Binomial Probability Formula

The binomial probability formula is a fundamental concept in probability theory, used to calculate the likelihood of observing a specific number of successes in a fixed number of independent trials, each with the same probability of success. This formula is essential when dealing with dichotomous outcomes – scenarios where there are only two possible results for each trial, such as success or failure.

In the context of our exercise, where we are looking to find the number of persons with type A blood in a sample, the formula is given by:
\(Pr(Y=k) = C(n, k) \times p^k \times (1-p)^{(n-k)}\), where:

• \(n\) is the number of trials,
• \(k\) is the number of successes,
• \(p\) is the probability of success,
• \(C(n, k)\) is the combination function representing the number of ways to choose \(k\) successes from \(n\) trials.

For our example, the probability of success (having type A blood) is 0.44. We plug this value, along with the number of trials (4) and the desired number of successes (0, 1, or 2), into the formula to calculate the probability for each case.

Cumulative Probability

Cumulative probability refers to the likelihood of observing a value within a certain range in a probability distribution. It is the sum of probabilities of all outcomes up to a specified value. In essence, it answers the question: 'What is the probability that the outcome will be at or below a particular value?'

In practice, we calculate cumulative probability by adding up the probabilities of all relevant outcomes. For example, to find the probability that there are at most two people with type A blood in our sample, we add the probabilities of having 0, 1, and 2 people with type A blood. Using the binomial probability formula from our previous section for each scenario, the cumulative probability for \(0 \text{≤} Y \text{≤} 2\) is the sum of \(Pr(Y=0)\), \(Pr(Y=1)\), and \(Pr(Y=2)\). This represents the total probability of observing between zero and two successes, inclusive.

Probability of Success

The probability of success is a term in probability theory that denotes the likelihood of a single trial resulting in a success. In a binomial distribution, each trial is independent, and the probability of success remains consistent across all trials.

The significance of this value is that it helps determine the skewness of the distribution; a lower probability of success leads to a distribution that is skewed to the right, while a higher probability of success leads to a distribution that is skewed to the left. In the provided exercise, the probability of success, denoted as \(p\), is 0.44. This means that in any given trial, there is a 44% chance that a randomly selected person has type A blood. Understanding this value is crucial as it is a key component of the binomial probability formula.

Number of Trials

The number of trials, often denoted as \(n\), in a binomial distribution represents how many times an experiment or process is carried out. The trials must be independent, meaning the outcome of one trial does not affect the others, and the probability of success should stay the same in each trial.

In our exercise, the number of trials is 4, which corresponds to the size of our sample. Determining the number of trials is important as it sets the stage for calculating the probabilities of different outcomes, and the total number of possible outcomes is determined by the power set of the number of trials. It is also used to calculate the combinations of successes in different scenarios, which is a critical component in the binomial probability formula.