91Ó°ÊÓ

Conditional probability is an important concept in probability theory that allows us to calculate the probability of one event happening, given that another event has already occurred. It's like asking, "What is the probability that event B happens if we already know event A has happened?" In this exercise, we are tasked with finding the probability of having a certain number of usable pints of blood under the condition that already at least 6 are usable.

This requires us to understand the formula for conditional probability: \[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]Here, \(P(A|B)\) is the probability of event A given event B.

- For part (a), we need to find \(P(8 \leq r | 6 \leq r)\). This means we need the probability of having at least 8 usable pints if we know we already have at least 6 usable pints. So, our event B is "at least 6 are usable," and event A is "at least 8 are usable."- For part (b), we compute \(P(r=10 | 6 \leq r)\). Here, event A is "all 10 pints are usable," while event B remains "at least 6 are usable."Using the given probabilities and the binomial distribution, we can calculate these probabilities effectively. The key is to first determine the probabilities of each event separately, then apply the formula to find the conditional probabilities.

The binomial theorem is vital not only in algebra but also in probability, especially when dealing with a binomial distribution. In our exercise, the scenario of determining usable blood pints follows a binomial distribution because each pint of blood is either usable or it isn’t, making it a sequence of independent binary outcomes.

A binomial distribution can be characterized by two parameters, \( n \), the number of trials (or pints of blood, in this case), and \( p \), the probability of success (usable pint). The binomial probability formula is given by:\[ P(r) = \binom{n}{r} p^r (1-p)^{n-r} \]Where:

• \( \binom{n}{r} \) is the binomial coefficient, "n choose r," indicating the number of combinations of \( n \) items taken \( r \) at a time.
• \( p^r \) is the probability of having exactly \( r \) successes, and \((1-p)^{n-r} \) accounts for the failures.

In our situation, \( n = 10 \) and \( p = 0.7 \), meaning each pint has a 70% chance of being usable. This formula helps us calculate the probability of any number of successes out of 10 pints being usable.

Calculating probabilities directly from the binomial distribution involves using the formula to determine the likelihood of a certain number of successes. In the exercise, we explored different probabilities, depending on the conditions set by using the binomial theorem.

First, we calculate the probability for a given \( r \) value using:\[ P(r) = \binom{10}{r} (0.7)^r (0.3)^{10-r} \]We use different values of \( r \) for each part:

• For \( P(r \geq 8) \), calculate \( P(8) + P(9) + P(10) \).
• For \( P(6 \leq r) \), compute \( 1 - P(r \leq 5) \).
• For \( P(r = 10) \), simply compute \( P(10) \).

Once these probabilities are calculated, they are used as inputs for the conditional probabilities \( P(8 \leq r | 6 \leq r) \) and \( P(r=10 | 6 \leq r) \) from the earlier section.

By doing so, we gain insights into the likely outcomes of the blood usability context, allowing us to understand the role of conditional probability in determining the usability of all blood pints."