Problem 69 - Basic Statistics for Business and Economics with Student

Question

A new chewing gum has been developed that is helpfu to those who want to stop smoking. If 60 percent of those people chewing the gum are successful in stopping smoking, what is the probability that in a group of four smokers using the gum at least one quits smoking?

Step-by-Step Solution

Answer

The probability that at least one person quits is 97.44%.

Step 1: Identify the Probability of Success

The probability of a single person successfully quitting smoking by using the gum is given as 60%. In probability terms, this is \( P(S) = 0.60 \).

Step 2: Calculate Probability of Failure

The probability that a person does not stop smoking is the complement of the probability they do. Therefore, \( P(F) = 1 - P(S) = 1 - 0.60 = 0.40 \).

Step 3: Determine Probability of All Failures

To find the probability that all four smokers fail to quit, we raise the probability of failure to the fourth power: \( P( ext{All Fail}) = (0.40)^4 \).

Step 4: Calculate Complement Probability

We seek the probability that at least one smoker quits, which is the complement of all four failing. Therefore, we subtract the probability of all failing from 1: \( P( ext{At least one quit}) = 1 - (0.40)^4 \).

Step 5: Compute the Result

Calculate \((0.40)^4\) to get 0.0256. Then, subtract from 1 to find the probability at least one quits: \( 1 - 0.0256 = 0.9744 \).

Key Concepts

Binomial Probability

When faced with scenarios involving yes-no outcomes, like the success or failure of an event, we tap into binomial probability. This type of probability helps us predict the likelihood of multiple successes (or failures) across a set number of trials.
For instance, let's consider a situation where each trial, like using our special gum, has the same success chance. In our chewing gum example, each smoker has a 60% chance of quitting, or \(0.60\).
Using the concept of binomial probability means considering:

• The fixed number of trials (here, four smokers).
• The consistent probability of success in each trial (60%, or \(0.60\)).
• The two potential outcomes per trial, success, or failure.

In such scenarios, the binomial formula can predict the probability of obtaining a certain number of successes out of the set trials. The formula is:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
Here, \(n\) is the number of trials, \(k\) is the number of successes, \(p\) is the probability of success, and \(\binom{n}{k}\) is the binomial coefficient, which calculates how many ways \(k\) successes can occur in \(n\) trials.

Complement Rule

In probability, sometimes it's easier to find the probability of an event not happening, and then use this to find the probability of the event happening. This is where the complement rule comes in handy.
The complement rule states that the probability of an event happening plus the probability of it not happening always equals one. Mathematically, if \(P(A)\) is the probability of event A, then the complement is given by:
\[ P(A^c) = 1 - P(A) \]
In our chewing gum scenario, we want to find the probability that at least one of the smokers successfully quits. Calculating this directly can be cumbersome. Instead, we find the probability that none quit (all fail) and subtract that from one to get our desired probability. This simplifies calculations when dealing with multiple trials and scenarios.
In this case, the probability that all four smokers fail is \((0.40)^4\), and therefore, the probability that at least one quits is:
\[ 1 - (0.40)^4 \]

Probability of Success

The 'probability of success' is the likelihood that a specific event will occur favorably. In our context of smoking cessation, it's the probability that a smoker using the gum will successfully quit.
This probability is often given or determined through studies or experiments. In our chewing gum case, it's provided as 60%, which we translate into probability terms as \( P(S) = 0.60 \).
When dealing with probabilities, remember:

• The probability of success and failure for an event always adds up to 1.
• Success can be different depending on the problem context; here, it's quitting smoking.

If you know the probability of success, you can easily determine the probability of failure by subtraction, \( P(F) = 1 - P(S) \).
Understanding these probabilities allows us to calculate more complex scenarios, like evaluating the likelihood of multiple successes or the overall chance at least one success occurs in a set of trials.