A new chewing gum has been developed that is helpful 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?
The only combination that would not include at least one person being successful is if all are not successful. If they have a .6 success rate, there is a .4 failure rate. Because all probabilities add up to 1.00.
P(person 1 failure) and P(person 2 failure) and P(person 3 failure) and P(person 4 failure)
And in probability means to multiply. What do we do with this?"
The probability of at least one quitting smoking is equal to 1-prob(all smoking)
Pr(all smoking)=.4^4
Pr(at least one quitting smoking)=1-.4^4
= greater than 96 percent (you do the calculation)
0.1041666
67
To calculate the probability of at least one person quitting smoking, we can use the complement rule. The complement of "at least one person quitting smoking" is "all people smoking."
Since each individual has a 0.4 (40%) chance of failure, the probability of all four people smoking (all failures) is calculated as follows:
Pr(all smoking) = 0.4^4 = 0.4 * 0.4 * 0.4 * 0.4 = 0.01696
Therefore, the probability of at least one person quitting smoking is:
Pr(at least one quitting smoking) = 1 – Pr(all smoking)
= 1 – 0.01696
= 0.98304
So, the probability that at least one person in the group of four smokers using the gum quits smoking is approximately 0.98304, or 98.3%.
To calculate the probability of at least one person quitting smoking in a group of four smokers using the gum, you need to find the complement of the probability that all four smokers continue smoking.
The probability of all four smokers continuing to smoke is calculated by multiplying the failure rate (0.4) for each individual smoker:
Pr(all smoking) = P(person 1 failure) * P(person 2 failure) * P(person 3 failure) * P(person 4 failure) = 0.4^4
The complement of this probability (1 - Pr(all smoking)) gives you the probability of at least one person quitting smoking:
Pr(at least one quitting smoking) = 1 - Pr(all smoking) = 1 - 0.4^4
You can then calculate the exact probability using a calculator or by simplifying the expression further.
So, in the given scenario, the probability of at least one person quitting smoking in a group of four smokers using the gum is greater than 96 percent.