Question

a random sampling of 4 members of a 150-member club has shown that 3 prefer no smoking in the clubhouse dining room. what is the probability that this will occur if in fact only 20% of members prefer no smoking in the dining room?

Answer 1

The probability that exactly 3 out of 4 randomly selected members prefer no smoking, when only 20% of the members prefer no smoking, can be calculated using the binomial probability formula and is found to be 2.56%.

Step-by-step explanation:

The question is asking to calculate the probability that if only 20% of the club members prefer no smoking in the dining room, a random sample of 4 members will have exactly 3 who prefer no smoking. To solve this, we use the binomial probability formula which is:

P(X=k) = C(n, k) × p^k × (1-p)^(n-k), where:

• C(n, k) is the combination of n items taken k at a time,
• p is the probability of a single event occurring,
• n is the number of events,
• and k is the number of times the event is to occur in those n events.

Given that p = 0.20 (the probability that a single member prefers no smoking), n = 4 (the number of members chosen), and k = 3 (the number that prefers no smoking), we can calculate the probability:

P(X=3) = C(4, 3) × (0.20)^3 × (0.80)^1

Using a calculator, we can determine that C(4, 3) = 4, so:

P(X=3) = 4 × 0.008 × 0.80

Now we multiply these values together to get the final probability:

Probability = 0.0256 or 2.56%

Thus, the probability that exactly 3 out of 4 randomly selected members prefer no smoking, given that 20% of the members prefer no smoking, is 2.56%.