Suppose that 20% of all individuals have an adverse reaction...

Suppose that 20$\%$ of all individuals have an adverse reaction to a particular drug. A medical researcher will administer the drug to one individual after another until the first adverse reaction occurs. Define an appropriate random variable and use its distribution to answer the following questions. (a) What is the probability that when the experiment terminates, four individuals have not had adverse reactions? (b) What is the probability that the drug is administered to exactly five individuals? (c) What is the probability that at most four individuals do not have an adverse reaction? (d) How many individuals would you expect to not have an adverse reaction, and how many $\quad$ individuals would you expect to be given the drug? (e) What is the probability that the number of individuals given the drug is within one standard deviation of what you expect?

Suppose that 20$\%$ of all individuals have an adverse reaction to a particular drug. A medical researcher will administer the drug to one individual after another until the first adverse reaction occurs. Define an appropriate random variable and use its distribution to answer the following questions.
(a) What is the probability that when the experiment terminates, four individuals have not had adverse reactions?
(b) What is the probability that the drug is administered to exactly five individuals?
(c) What is the probability that at most four individuals do not have an adverse reaction?
(d) How many individuals would you expect to not have an adverse reaction, and how many $\quad$ individuals would you expect to be given the drug?
(e) What is the probability that the number of individuals given the drug is within one standard deviation of what you expect?

Key Concepts

Expected Value

The expected value of a geometric distribution is the mean number of trials needed to achieve the first success, calculated as E(X) = 1/p. This parameter provides a measure of the central tendency or average performance of repeated experiments and is fundamental in understanding the long-term behavior of the process.

Variance and Standard Deviation

The variance of a geometric distribution, given by Var(X) = (1 - p)/p^2, measures the spread or variability around the expected value. The standard deviation, which is the square root of the variance, indicates the typical amount by which the number of trials may deviate from the mean. These measures are crucial for assessing the uncertainty and dispersion inherent in the process of repeated independent trials.

Cumulative Distribution Function

The cumulative distribution function (CDF) for a geometric distribution provides the probability that the first success occurs on or before a given trial. It is obtained by summing the PMF over all applicable trials. This is useful for determining the likelihood that an event happens within a certain number of attempts.

Probability Mass Function

The probability mass function (PMF) for a geometric distribution gives the probability that the first success occurs on the kth trial. It is expressed as P(X = k) = (1 - p)^(k - 1) * p, where p is the probability of success on any individual trial. This function is used to calculate probabilities for specific outcomes in sequential experiments.

Geometric Distribution

The geometric distribution models the number of independent, identical trials needed to achieve the first success in a sequence of Bernoulli trials. It is characterized by a constant probability of success on each trial, and it assumes that trials are independent of each other. This distribution is fundamental in scenarios where an experiment is repeated until a particular event occurs for the first time.

Transcript

00:01 In this question, we are told that 20 % of individuals have an adverse reaction to a particular drug.

00:06 So we can see that probability is 0 .2.

00:11 And the drug is administered to individuals one after another until the first adverse reaction.

00:19 So we're looking at the number of trials until we have exactly one success.

00:26 And we are asked to define an appropriate random variable and use its distribution to answer the questions.

00:32 And the questions a through e all pertain to how many trials must occur before we have one adverse reaction.

00:43 So we can define our random variable, x, as the number of drug administrations until the first adverse reaction.

01:00 So if you're thinking this is a negative binomial distribution, that is right.

01:05 It's also a geometric random variable, and that's just a special case of the negative binomial, when r is equal to 1.

01:13 So for a geometric distribution, the probability mass function is simply 1 minus p to the exponent x minus 1 times p, and that is for x is a positive integer.

01:43 For part a we were asked the probability that when the experiment terminates that four individuals have not had adverse reactions.

01:51 So that means we have the first four individuals do not have a reaction, and then the fifth one has a reaction, and that's when it terminates.

02:00 So this can only be the probability that we have five trials.

02:08 That's four no reactions followed by one adverse reaction.

02:18 So it's 0 .8, which is 1 minus p.

02:28 And this probability comes out to 0 .082.

02:36 In part b, we are looking for the probability that the drug is administered to exactly five individuals.

02:43 So this is the same question as part a, just worded, in a different way.

02:48 If you have four non -adverse reactions followed by one adverse reaction, that is the same as having the drug administered to a total of five people.

03:09 For part c, we are asked the probability that at most four individuals do not have an adverse reaction.

03:16 That is the same as the probability that at most five drug administrations take place before it ends.

03:26 So if we have at most five people being administered the drug, we know that at most people being administered the drug, we know that at most most 4 have had no reaction...

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