Question

A certain drug treatment cures \(90 \%\) of cases of hookworm in children. \({ }^{18}\) Suppose that 20 children suffering from hookworm are to be treated, and that the children can be regarded as a random sample from the population. Find the probability that (a) all 20 will be cured. (b) all but 1 will be cured. (c) exactly 18 will be cured. (d) exactly \(90 \%\) will be cured.

Short answer

The probabilities are: (a) all 20 will be cured is 0.9^20; (b) all but 1 will be cured is 20 * 0.9^19 * 0.1; (c) exactly 18 will be cured is C(20,18) * 0.9^18 * 0.1^2; (d) the probability exactly 90% (18 out of 20) will be cured is the same as part (c).

Step-by-step solution

Understanding The Problem

This is a binomial probability problem where the success rate (curing hookworm in children) is 90%. The total number of trials (children) is 20. We need to find different probabilities based on the binomial distribution.

Setting Up The Binomial Probability Formula

The binomial probability formula is given by P(X = k) = C(n,k) * p^k * (1-p)^(n-k), where P(X = k) is the probability of k successes in n trials, p is the success probability, and C(n,k) is the combination of n items taken k at a time.

Finding the Probability All 20 Children Are Cured (a)

Using the formula for k = 20 (all children are cured): P(X = 20) = C(20,20) * (0.9)^20 * (0.1)^0 = 1 * 0.9^20 * 1

Calculating Probability for Exact Outcomes

Perform the calculation from the formula in Step 3: P(X = 20) = 0.9^20.

Finding the Probability All but 1 Child is Cured (b)

Using the formula for k = 19: P(X = 19) = C(20,19) * (0.9)^19 * (0.1)^1.

Calculating Probability for All but 1 Child Cured

Perform the calculation from the formula in Step 5: P(X = 19) = 20 * 0.9^19 * 0.1.

Finding the Probability Exactly 18 Children Are Cured (c)

Using the formula for k = 18: P(X = 18) = C(20,18) * (0.9)^18 * (0.1)^2.

Calculating Probability for Exactly 18 Children Cured

Perform the calculation from the formula in Step 7: P(X = 18) = C(20,18) * 0.9^18 * 0.1^2.

Finding the Probability Exactly 90% of the Children Are Cured (d)

Since 90% of 20 children is 18, this is the same as part (c). Thus, P(X = 18) is the answer.

Key concepts

Probability Distribution

In our daily life, we often encounter situations where the outcomes are uncertain. To analyze these situations, we use probability distributions, which are mathematical functions that provide the probabilities of occurrence of different possible outcomes. They not only tell us what outcomes can happen but also how likely they are.

There are many types of probability distributions, and one important type is the binomial probability distribution. This distribution is used when an experiment, known as a Bernoulli trial, is repeated a fixed number of times. Each trial has only two possible outcomes, typically referred to as 'success' and 'failure'. The binomial distribution allows us to calculate the probability of obtaining a certain number of successes in a series of trials. In the context of the exercise, curing hookworm in children is considered a 'success', while not curing it is a 'failure', with each child being an independent trial.

Binomial Theorem

The binomial theorem provides a powerful shortcut for expanding algebraic expressions raised to a power. It states that for any positive integer n, the nth power of the sum of two numbers a and b can be expanded as follows:
\[(a + b)^n = \sum_{k=0}^{n} C(n,k) a^{n-k} b^k.\]

Here, C(n,k) represents the binomial coefficient, often read as 'n choose k', which is the number of ways to choose k items from a set of n distinct items. In our exercise, the binomial theorem underlies the formula used to calculate probabilities in the binomial distribution. Understanding how to apply this theorem is crucial in solving problems involving repeated trials with two outcomes.

Combinatorics

Combinatorics is a field of mathematics primarily concerned with counting, ordering, and arranging objects. One key concept in combinatorics is the calculation of combinations, which are used to determine the number of ways 'k' objects can be chosen from a set of 'n' objects without considering the order. This is represented by the binomial coefficient C(n,k) introduced earlier.

When solving the given binomial probability problem, we use combinatorial calculations to find out the number of ways a certain number of children can be cured. Specifically, C(20,20), C(20,19), and C(20,18) calculate the number of ways all children, all but one child, and exactly 18 children can be cured, respectively. Understanding combinatorics helps us in figuring out these key components of our probability problem.

Success Probability

In probability theory, the success probability is the chance that a single trial results in a success. This is a fundamental concept in the study of binomial distributions, as the distribution parameters are determined by the number of trials and the success probability.

For our exercise, the success probability is given as 90%, or 0.9, for each child to be cured of hookworm. This probability is essential to calculate the probability of any number of children being cured out of the 20 treated. Knowledge of success probability allows us to understand the dynamics of the binomial formula and to predict outcomes for different scenarios.