Drug Effectiveness A new drug cures 80% of the patients to...

Drug Effectiveness A new drug cures 80$\%$ of the patients to whom it is administered. It is given to 25 patients. Find the probabilities that among these patients, the following results occur. a. Exactly 20 are cured. b. All are cured, c. No one is cured. d. Twelve or fewer are cured.

Drug Effectiveness A new drug cures 80$\%$ of the patients to whom it is administered. It is given to 25 patients. Find the probabilities that among these patients, the following results occur.
a. Exactly 20 are cured.
b. All are cured,
c. No one is cured.
d. Twelve or fewer are cured.

Key Concepts

Binomial Distribution

The binomial distribution is a discrete probability distribution that expresses the probability of a given number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by the number of trials (n) and the probability of success (p) in each trial, and is used to calculate probabilities of outcomes like exactly k successes, all successes, or no successes.

Probability Mass Function (PMF)

The PMF of a binomial distribution gives the probability of obtaining exactly k successes in n independent trials. It is defined by the formula P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) represents the binomial coefficient. This formula is essential for calculating the probabilities of specific outcomes in binomial experiments.

Binomial Coefficient

The binomial coefficient, denoted as C(n, k) or 'n choose k', represents the number of distinct ways to choose k successes out of n independent trials. It plays a crucial role in the probability mass function of the binomial distribution, as it quantifies the different combinations in which the k successes can occur.

Cumulative Distribution Function (CDF)

The cumulative distribution function for the binomial distribution gives the probability that the number of successes is less than or equal to a certain value. It is used to determine probabilities over a range of outcomes, such as finding the probability of having twelve or fewer successes. The CDF is obtained by summing the PMF values for all outcomes up to the specified number.

Transcript

00:01 In this problem, we're told of a new drug that is set to cure 80 % of the patients that it's administered to.

00:10 And if we have 25 patients, we're asked to find the following probabilities.

00:17 So let's note the important information.

00:20 We know that p is equal to 0 .8.

00:23 And we know that n is equal to 25.

00:27 Since we will be using the normal approximation, we will need to find the mean and the standard deviation.

00:36 So the mean is given by n times p, which is equal to 20.

00:48 And our standard deviation is given by the square root of 25 times p times 1 minus p.

01:04 And that is equal to 2.

01:06 So in part a where we asked to find exactly the probability of exactly 20 people cured, that's the probability where x is equal to 20.

01:20 We know that corresponds to the area beneath the bar of the histogram from x equal 19 .5 to x equal 20 .5.

01:40 So we need to find the corresponding z scores for those to x values.

01:46 So the first one we get by doing 19 .5 minus the mean divided by the standard deviation.

01:57 This works out to be negative 0 .25.

02:07 And the second z score 20 .5 minus 20 over 2 works out to be positive.

02:29 Therefore, the probability that we have exactly 20 people cured is they go to the probability that z is less than 0 .25 minus the probability that z is less than negative 0 .25.

02:52 From the z table, we get 5987 minus 0 .4013 works out to a final probability of approximately 0 .1974.

03:15 In part b, we're asked to find the probability that all are cured.

03:40 And basically we're finding the probability that x is equal to 25.

03:52 And we know that corresponds to the area under the bar of our histogram from x equal 24 .5 to x equal 25 .5.

04:19 The corresponding z scores, 24 .5 minus 20 over 2, gives approximately 2 .75, or 2 .25 rather...

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