Question

Only \(4 \%\) of people have Type AB blood. a) On average, how many donors must be checked to find someone with Type \(\mathrm{AB}\) blood? b) What's the probability that there is a Type \(\mathrm{AB}\) donor among the first 5 people checked? c) What's the probability that the first Type \(\mathrm{AB}\) donor will be found among the first 6 people? d) What's the probability that we won't find a Type \(\mathrm{AB}\) donor before the 10 th person?

Short answer

a) 25 donors. b) 18.5%. c) 21.77%. d) 66.48%.

Step-by-step solution

Determine the Probability of a Type AB Donor

The probability of a donor having Type AB blood is given as \( p = 0.04 \). This means that \( 4\% \) of the population is expected to have Type AB blood.

Calculate the Average Number of Donors Checked (Part a)

The average number of donors that need to be checked to find someone with Type AB blood can be modeled using the concept of 'expected value'. The expected number of trials to get the first success in a geometric distribution is given by \( \frac{1}{p} \). Thus, we calculate \( \frac{1}{0.04} = 25 \). On average, 25 donors must be checked.

Calculate the Probability Among First 5 (Part b)

To find the probability that there is at least one Type AB donor in the first 5 donors, we use the complement rule. First, find the probability that none of the first 5 donors have Type AB blood: \( (1 - p)^5 = 0.96^5 \). Thus, the probability of at least one is \( 1 - 0.96^5 \). This calculates to about \( 0.185 \) or \( 18.5\% \).

Determine Probability of First 6 Trials (Part c)

The probability that the first Type AB donor is found within the first 6 donors is \( 1 - (1 - p)^6 = 1 - 0.96^6 \), which is about \( 0.2177 \) or \( 21.77\% \).

Calculate Probability No AB until 10th (Part d)

The probability that we do not find a Type AB donor before the 10th person means all first 9 donors do not have Type AB blood. This is \( (1-p)^9 = 0.96^9 \), which calculates to approximately \( 0.6648 \) or \( 66.48\% \).

Key concepts

Probability Theory

Probability theory is a mathematical framework that helps us understand randomness and uncertainty in various situations. It provides the tools to quantify the likelihood of different outcomes. Using probability, we can make informed guesses about the occurrence of events.

For instance, if you know that the probability of a donor having Type AB blood is 4%, you can predict how often this blood type will appear in a group of donors. In probability theory, these chances are usually represented by the symbol \( p \).

In our exercise, \( p = 0.04 \), indicating a 4% chance. Probability is essential for calculating the likelihood of events, such as finding a Type AB blood donor among a group of potential donors.

• Probabilities range from 0 to 1.
• A probability of 1 means an event is certain to happen.
• A probability of 0 means an event is impossible.

Understanding the basics of probability can aid us in various real-world scenarios, including making decisions under uncertainty.

Expected Value

The expected value is a key concept in probability and statistics, providing a measure of the center of a probability distribution. In simpler terms, it's like an average outcome you can expect from a random event.

In the context of our blood donor example, if we want to know how many donors we need to check on average to find someone with Type AB blood, we calculate the expected value. When dealing with a geometric distribution, which models the number of trials until the first success, the formula is \( \frac{1}{p} \).

For this exercise, with \( p = 0.04 \), the expected number of donors is \( \frac{1}{0.04} = 25 \). This means that on average, 25 donors need to be checked to find someone with Type AB blood.

• The expected value helps predict long-term average outcomes.
• It simplifies the decision-making process in uncertain situations.

Using the expected value can give us a clear picture of what to expect in repeated or similar situations.

Complement Rule

The complement rule is a handy tool in probability, especially when it's easier to calculate the likelihood of an event not happening rather than happening directly. The rule states that the probability of an event occurring is \( 1 \) minus the probability of it not occurring.

Applying this rule is crucial in problems like ours. To find the probability of at least one Type AB donor among the first 5 people checked, we first calculate the probability that none of them has Type AB blood. This is given by \( (1-p)^5 = 0.96^5 \).

Then using the complement rule, we find \( 1 - (1-p)^5 = 1 - 0.96^5 \), which equates to approximately 18.5%. This gives us the likelihood of our desired scenario happening.

• The complement rule simplifies complex probability calculations.
• It is particularly useful when dealing with multiple trials or events.
• The sum of the probability of an event and its complement equals 1.

The complement rule is an essential technique for tackling probability questions efficiently and correctly.