(1) The director of a health club conducted a survey and found that 23% of members
used only the pool for workouts. Based on this information, what is the probability that for a random sample of 10 members, 4 used only the pool for workouts? Round your answer to four decimal places.
To find the probability that 4 out of 10 members use only the pool for workouts, we need to use the binomial probability formula. The formula is given by:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
P(x) is the probability of getting exactly x successes
n is the total number of trials (sample size)
x is the number of successful trials (number of members using only the pool for workouts)
p is the probability of success for each individual trial (23% or 0.23)
(1-p) is the probability of failure for each individual trial (77% or 0.77)
nCx is the number of combinations of n items taken x at a time, which can be calculated using nCx = n! / (x! * (n-x)!)
Let's plug in the values for our problem:
n = 10 (total number of members in the random sample)
x = 4 (number of members using only the pool for workouts)
p = 0.23 (probability of each member using only the pool for workouts)
(1-p) = 1 - 0.23 = 0.77 (probability of each member not using the pool for workouts)
Now we can calculate:
P(4) = (10C4) * (0.23)^4 * (0.77)^(10-4)
To calculate (10C4), we use the combination formula:
10C4 = 10! / (4! * (10-4)!)
= 10! / (4! * 6!)
= (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1)
= 210
Now substitute the values back into the formula:
P(4) = 210 * (0.23)^4 * (0.77)^6
Calculating this expression will give you the probability that exactly 4 out of 10 members use only the pool for workouts. Round the answer to four decimal places as requested.