A survey revealed that 44% of movie goers will eat some amount of popcorn during a movie. If the next 17 movie goers are asked if they will eat popcorn during the next movie they attend, what is the probability that at least 8 will say they will eat popcorn during the next movie?
p eat = .44
p not eat = .56
p that k say yes (binomial)
P(17/k) = C(17,k) .44^k .56(17-k)
where
C(17,k) = 17!/[k!(17-k)!]
do that for k = 8 through 17 and add
OR
approximate binomial expansion with normal distribution
mean = n p = .44 * 17 = 7.48
sigma^2 = 17(.44)(.56) = 4.19 = n p(1-p)
use david lane tables http://davidmlane.com/hyperstat/z_table.html
First write out what you know in order to get the formula:
x = # of movie goers
44% of movie goers eat popcorn
so 44% of movie goers = movie goers who eat popcorn
Out of 17 movie goers how many will eat popcorn?
44% of x =
44% of 17 = 7 movie goers who will eat popcorn
But the question is asking for the percentage if the # of movie goers who eat popcorn = 8 so you know the answer has to be higher than 44%
The formula becomes:
y% of 17 = 8
y% x 17 = 8
y% = 8/17
y = (8/17)*100 = 47.06%
First, 47.06% is higher than 44% (logical verification)
Second, try it out to prove it:
47.06% x 17 = 8.0002
To find the probability that at least 8 movie-goers out of 17 will say they will eat popcorn during the next movie, we need to use the binomial probability formula.
The formula for binomial probability is:
P(X=k) = (n C k) * p^k * (1-p)^(n-k)
Where:
P(X=k) is the probability of getting exactly k successes.
n is the number of trials.
k is the number of successes.
p is the probability of success in a single trial.
In this case, we want to find the probability of getting at least 8 successes out of 17 trials, with a success rate of 44%.
Step 1: Calculate the probability of getting exactly 8, 9, 10, 11, ... up to 17 successes.
P(X=8) = (17 C 8) * (0.44)^8 * (0.56)^(17-8)
P(X=9) = (17 C 9) * (0.44)^9 * (0.56)^(17-9)
P(X=10) = (17 C 10) * (0.44)^10 * (0.56)^(17-10)
...
P(X=17) = (17 C 17) * (0.44)^17 * (0.56)^(17-17)
Step 2: Sum up all the probabilities calculated in Step 1 to get the probability of getting at least 8 successes.
P(X>=8) = P(X=8) + P(X=9) + P(X=10) + ... + P(X=17)
Note: The combination (n C k) can be calculated using the formula:
(n C k) = n! / (k! * (n-k)!)
You can now plug in the values and calculate the probabilities using a calculator or a software program.