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Chapter 5: Problem 107

A doctor knows from experience that \(10 \%\) of the patients to whom she gives a certain drug will have undesirable side effects. Find the probabilities that among the 10 patients to whom she gives the drug: a. At most two will have undesirable side effects. b. At least two will have undesirable side effects.

Short Answer

Expert verified
The probability that at most two patients will have undesirable side effects is .9298 or 92.98%. And the probability that at least two patients will have undesirable side effects is .2639 or 26.39%.

Step by step solution

01

Calculating the probability of at most two patients having side effects

The first part involves calculating the probability of at most two patients out of ten patients having side effects. The keyword 'at most' means we are considering the cases where zero, one or two patients have side effects. Since these are discrete events, each probability is calculated separately using binomial distribution formula and then summed up to find the total probability:i) \( P(X = 0) = {10 \choose 0}(.1)^0(.9)^{10-0} = .3487 \)ii) \( P(X = 1) = {10 \choose 1}(.1)^1(.9)^{10-1} = .3874 \)iii) \( P(X = 2) = {10 \choose 2}(.1)^2(.9)^{10-2} = .1937 \)'At most two patients will have side effects' can be calculated by summing these three probabilities.
02

Summing up the probabilities

As we have calculated these probabilities separately we will now sum them up : \( P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = .3487 + .3874 + .1937 = .9298 \)So the probability that at most two patients will have undesirable side effects is .9298 or 92.98%.
03

Calculating the probability of at least two patients having side effects

The second part involves calculating the probability of at least two patients out of ten patients having side effects. The keyword 'at least' means to find the probability of two or more patients having side effects. The best way to approach 'at least' problems with binomial distribution is to think about what is the complement of the event (i.e., think about what we don't want). In other words, the complement to 'at least two have side effects' is 'one or none have side effects'. We already know the probability that one or none have side effects from the first part of the problem (P(X = 0) and P(X = 1)). So we can simply subtract the sum of these probabilities from 1 to get the desired probability: \( P(X \geq 2) = 1 - (P(X = 0) + P(X = 1)) = 1 - (.3487 + .3874) \)
04

Finding the final probability

After substituting the above values in the formula we will get the final probability. \( P(X \geq 2) = 1 - (.3487 + .3874) = .2639 \)This implies that the probability that at least two patients will have undesirable side effects is .2639 or 26.39%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory
At the heart of statistics and gambling lies probability theory, the branch of mathematics that measures the likelihood of an event occurring. It is the foundation upon which random events are analyzed, providing a way to express uncertainty numerically.

Within this realm, we can consider events like coin tosses, dice rolls, or in our case, the occurrence of side effects in patients receiving a certain drug. Probability theory gives us tools and formulas to calculate these chances, and it does so by using different types of probability distributions that model various scenarios.

When dealing with probability, we have a few basic rules. The probability of an impossible event is 0, while a certain event has a probability of 1. Moreover, the sum of probabilities of all possible outcomes of a discrete random variable is always 1. These foundational concepts lead us to solve more complex problems, such as determining the probability that at most or at least a certain number of patients experience side effects from a medication.
Discrete Random Variables
A discrete random variable is a type of variable that can take on a countable number of distinct values. Each value corresponds to an outcome or a combination of outcomes from a statistical experiment. In our example, the discrete random variable can be defined as the number of patients experiencing side effects after drug administration.

In the context of discrete random variables, we often use probability mass functions (PMF) to define the probability of each possible value the variable can take. The calculation, as demonstrated in the exercise, shows how various probabilities may be assigned to the number of patients affected by the drug's side effects, ranging from zero to ten. Understanding discrete random variables is crucial because they enable us to use specific probability distributions, such as the binomial distribution, to solve real-world problems involving discrete, countable outcomes.
Binomial Probability
The binomial distribution is a discrete probability distribution that applies to scenarios where there are a fixed number of independent trials, each with the same probability of a particular outcome. Each trial can result in just two outcomes 鈥� commonly referred to as 'success' or 'failure'.

In our exercise, 'success' could be defined as a patient experiencing side effects from the drug, while 'failure' is the absence of side effects. The binomial probability formula incorporates the number of trials (in this case, 10 patients), the probability of success for each trial (10% chance of side effects), and the specific number of successes one is solving for (e.g., no more than two patients experiencing side effects).

Understanding this concept is pivotal, as it enables us to calculate the probability of 'at most' and 'at least' events through a series of binomial probabilities. By summing these probabilities or using complements (1 minus the probability of the opposite event), students can solve a wide array of practical problems, encompassing tests of medication to quality control in manufacturing.

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Most popular questions from this chapter

A social worker is involved in a study about family structure. She obtains information regarding the number of children per family in a certain community from the census data. Identify the random variable of interest, determine whether it is discrete or continuous, and list its possible values.

Use a computer to find the probabilities for all possible \(x\) values for a binomial experiment where \(n=30\) and \(p=0.35.\)

The number of ships to arrive at a harbor on any given day is a random variable represented by \(x .\) The probability distribution for \(x\) is as follows: $$\begin{array}{l|lllll}\hline \boldsymbol{x} & 10 & 11 & 12 & 13 & 14 \\\\\boldsymbol{P}(\boldsymbol{x}) & 0.4 & 0.2 & 0.2 & 0.1 & 0.1 \\\\\hline\end{array}$$ Find the probability of the following for any a given day: a. Exactly 14 ships arrive. b. At least 12 ships arrive. c. At most 11 ships arrive.

\(\mathrm{A}\) die is rolled 20 times, and the number of "fives" that occur is reported as being the random variable. Explain why \(x\) is a binomial random variable.

In another germination experiment involving old seed, 50 rows of seeds were planted. The number of seeds germinating in each row were recorded in the following table (each row contained the same number of seeds). $$\begin{array}{cc|cc}\begin{array}{c}\text { Number } \\\\\text { Germinating }\end{array} & \begin{array}{c}\text { Number } \\\\\text { of Rows }\end{array} & \begin{array}{c}\text { Number } \\\\\text { Germinating }\end{array} & \begin{array}{c}\text { Number } \\\\\text { of Rows }\end{array} \\\\\hline 0 & 17 & 3 & 2 \\\1 & 20 & 4 & 1 \\\2 & 10 & 5 \text { or more } & 0 \\\\\hline\end{array}$$ a. What probability distribution (or function) would be helpful in modeling the variable "number of seeds germinating per row"? Justify your choice. b. What information is needed in order to apply the probability distribution you chose in part a? c. Based on the information you do have, what is the highest or lowest rate of germination that you can estimate for these seeds? Explain.
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