SOLUTION: 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 t
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Question 329231: 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?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
The hard way to do this is to calculate the probability that exactly 1 is successful and add that to the probability that exactly 2 are successful plus the probability that exactly 3 are successful plus the probability that exactly 4 are successful. The probability of exactly 
successes in 
trials where 
is the probability of success on any given trial is given by:
\ =\ \left(n\cr k\right\)p^k\left(1\,-\,p\right)^{n\,-\,k})
Where )
is the number of combinations of things taken at a time and is calculated by !})
So, in order to find the probability of at least 1 of 4, 
, you need:
\ =\ \sum_{i\,=\,1}^4\ \left(4\cr i\right\)(0.6)^i\left(0.4\right)^{4\,-\,i})
Which is a calculation with all the entertainment value of watching paint dry. Fortunately, there is an easier way.
The probability that at least one of anything happening is 1 minus the probability that none happens.
The probability that you achieve at least one success from four trials is then:
\ =\ 1\ -\ P_4(0,0.6)\ =\ 1\ -\ \left(4\cr 0\right\)(0.6)^0\left(0.4\right)^{4})
Which is a MUCH simpler calculation, especially since \ =\ 1)
and 
So this all reduces to:
\ =\ 1\ -\ (0.4)^4\ =\ 0.9744\ \approx\ 97%)
My calculator said it, I believe it, that settles it.
John
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