The director of a health club conducted a survey and found that 23% of memebers used only the pool for workouts. Based on this information, what is the probablity that for a random sample of 10 members, 4 used only the pool for workouts?
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Thank you
To find the probability that exactly 4 out of 10 members used only the pool for workouts, we can use the binomial probability formula:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Where:
P(X = k) represents the probability of getting exactly k successes
n represents the total number of trials
k represents the number of successes
p represents the probability of success in each trial
(1 - p) represents the probability of failure in each trial
In this case, n = 10, k = 4, p = 0.23, and (1 - p) = 0.77.
So the probability is calculated as follows:
P(X = 4) = (10C4) * (0.23)^4 * (0.77)^(10-4)
Now, let's compute each part separately:
(10C4) = 10! / (4! * (10-4)!) = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1) = 210
(0.23)^4 ≈ 0.00259
(0.77)^(10-4) = (0.77)^6 ≈ 0.241
Now, substitute these values back into the formula:
P(X = 4) = 210 * 0.00259 * 0.241 ≈ 0.121
Hence, the probability that for a random sample of 10 members, exactly 4 used only the pool for workouts is approximately 0.121, or 12.1%.