Problem 29 - Algebra and Trigonometry

Question

Pharmaceuticals \(A\) drug that is used to prevent motion sickness is found to be effective about 75\(\%\) of the time. Six friends, prone to seasickness, go on a sailing cruise, and all take the drug. Find the probability of each event. (a) None of the friends gets seasick. (b) All of the friends get seasick. (c) Exactly three get seasick. (d) At least two get seasick.

Step-by-Step Solution

Answer

(a) 0.178; (b) 0.000244; (c) 0.132; (d) 0.467.

Step 1: Understanding the Problem

We are dealing with a binomial probability problem. The probability of the drug being effective (preventing seasickness) is 75%, or 0.75. Thus, the probability of a person still getting seasick is 1 - 0.75 = 0.25. We have 6 friends, and we need to find probabilities for different scenarios based on these probabilities.

Step 1: Probability None Get Seasick

To find the probability that none of the friends gets seasick, we use the binomial probability formula with 0 successes (0 getting seasick). This equals: \[ P(X = 0) = \binom{6}{0} (0.25)^0 (0.75)^6 = 1 \times 1 \times 0.75^6 \]Calculate \(0.75^6\) to find this probability.

Step 3: Calculating None Get Seasick

\(0.75^6 = 0.178\). So, the probability that none of the friends gets seasick is \(0.178\).

Step 2: Probability All Get Seasick

To find the probability that all friends get seasick, use the formula with 6 successes (6 getting seasick): \[ P(X = 6) = \binom{6}{6} (0.25)^6 (0.75)^0 = 1 \times (0.25)^6 \times 1 \]Calculate \(0.25^6\) to find this probability.

Step 5: Calculating All Get Seasick

\(0.25^6 = 0.000244\). So, the probability that all of the friends get seasick is \(0.000244\).

Step 3: Probability Exactly Three Get Seasick

For exactly 3 friends getting seasick, we use the binomial probability with 3 successes:\[ P(X = 3) = \binom{6}{3} (0.25)^3 (0.75)^3 \]\(\binom{6}{3}\) is the binomial coefficient which can be calculated as 20. Then solve the equation.

Step 7: Calculating Exactly Three Get Seasick

\( (0.25)^3 = 0.015625 \) and \( (0.75)^3 = 0.421875 \). Hence, \[ P(X = 3) = 20 \times 0.015625 \times 0.421875 = 0.131836 \].

Step 4: Probability At Least Two Get Seasick

To find the probability that at least 2 friends get seasick, we calculate 1 minus the probability of less than 2 getting seasick, which is 0 or 1. We already have the probabilities for 0: \( P(X = 0) \) and now calculate for 1: \( P(X = 1) \).

Step 9: Calculating At Least Two Get Seasick

Calculate \( P(X = 1) = \binom{6}{1} (0.25)^1 (0.75)^5 = 6 \times 0.25 \times 0.237 = 0.355 \). Now, \( P(X < 2) = P(X = 0) + P(X = 1) = 0.178 + 0.355 = 0.533 \). Therefore, \( P(X \geq 2) = 1 - 0.533 = 0.467 \).

Step 10: Summary of the Probabilities

(a) Probability none gets seasick: 0.178 (b) Probability all get seasick: 0.000244 (c) Probability exactly three get seasick: 0.132 (d) Probability at least two get seasick: 0.467.

Key Concepts

Probability Calculation

Probability calculation is a key concept in determining the likelihood of various outcomes in different scenarios. For example, in the context of motion sickness prevention with a drug that is effective 75% of the time, we use probability to determine how likely it is for a certain number of friends to experience seasickness. In the binomial probability setting, we calculate probabilities by considering all possible outcomes and their respective chances.When calculating these probabilities, it's essential to understand the probability of success (drug being effective) and probability of failure (drug being ineffective):- Probability of success (\(p\)) = 0.75.- Probability of failure (\(q\)) = 1 - 0.75 = 0.25.
Calculating scenarios, such as all friends getting seasick or none, involves applying the correct probability values and multiplying through the appropriate number of occurrences, as dictated by your situation.

Binomial Coefficient

The binomial coefficient plays a crucial role in probability calculations for binomial distributions.It represents the number of ways of picking a certain number of successes out of a fixed number of trials.In mathematical terms, it is expressed as the number of combinations: \(\binom{n}{k}\ = \frac{n!}{k!(n-k)!}\), where \(n\) is the total number of trials, and \(k\) is the number of successes you are calculating for.For instance, when calculating the probability that exactly three out of six friends get seasick, the coefficient \(\binom{6}{3}\) represents the number of ways to choose 3 instances of seasickness out of 6 trials.This is calculated as 20 (computed as \(\frac{6\times5\times4}{3\times2\times1}\)). This value is then multiplied by the probabilities for each friend and scenario to establish the likelihood of the event.

Motion Sickness Prevention

Motion sickness prevention often involves understanding how effective a treatment is under given circumstances. For this example, the drug taken by the friends is effective 75% of the time, which serves as a crucial statistic for all calculations. Understanding this effectiveness allows us to estimate the risks and predict how effective treatment will be for a group. By using this data in conjunction with binomial probability distributions, we can predict different outcomes like: - Nobody gets sick - Everybody gets sick - A certain number of friends get sick (like exactly 3 or at least 2).
Such problems not only support practical decision making in health-related scenarios but also highlight the mathematical precision needed to evaluate effectiveness probabilities in real-world situations.

Binomial Probability Distribution

The binomial probability distribution is a fundamental concept in statistics that deals with the probability of a series of independent events that have two possible outcomes. In this context of motion sickness prevention, the drug effectiveness scenario fits this model perfectly.With binomial distribution, we evaluate the probability of achieving a specific number of successful outcomes over a series of trials. The formula for calculating each probability is \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Here, \(P(X = k)\) represents the probability of exactly \(k\) successes,\(n\)is the number of trials, \(k\) is the number of successful outcomes, p is the probability of success on each trial, and \(1-p\) is the probability of failure on each trial.Understanding and being able to use binomial probability distributions enables us to forecast and prepare for potential real-world outcomes, like predicting how likely it is that at least two friends will get seasick even after using an effective drug.