a marketing manager at Game superstore has made a claim that a customer would prefer deluxe model to the standard model of a product is 30%.
a)what is the probability that exactly four out of 10 customers will prefer the deluxe model?
b)what is the probability that more than three out of 10 customers will prefer the standard model?
c)what is the probability that at most two out of 10 customers will prefer the deluxe model?
a) To find the probability that exactly four out of 10 customers will prefer the deluxe model, we can use the binomial probability formula: P(X=k) = C(n, k) * p^k * (1-p)^(n-k)
Where:
- n = number of trials (in this case, 10 customers)
- k = number of successes (in this case, 4 customers preferring deluxe)
- p = probability of success (in this case, 30% or 0.3)
- C(n, k) = combination of n things taken k at a time, which can be calculated as C(n, k) = n! / (k!(n-k)!)
Plugging in the values, we get:
P(X=4) = C(10, 4) * 0.3^4 * (1-0.3)^(10-4)
P(X=4) = (10! / (4!(10-4)!)) * 0.3^4 * 0.7^6
P(X=4) ≈ 0.2001
So the probability that exactly four out of 10 customers will prefer the deluxe model is approximately 20.01%.
b) The probability that more than three out of 10 customers will prefer the standard model is the complement of the probability that three or fewer customers will prefer the deluxe model (since if more than three prefer the standard model, that means three or fewer prefer the deluxe model).
We can find the probability that three or fewer customers will prefer the deluxe model by summing the probabilities for 0, 1, 2, and 3 customers:
P(X≤3) = P(X=0) + P(X=1) + P(X=2) + P(X=3)
Calculating those individual probabilities using the same formula as before, we get:
P(X≤3) ≈ 0.0282 + 0.1211 + 0.2335 + 0.2668
P(X≤3) ≈ 0.6496
So the probability that more than three out of 10 customers will prefer the standard model is approximately 64.96%.
c) To find the probability that at most two out of 10 customers will prefer the deluxe model, we can sum the probabilities for 0, 1, and 2 customers:
P(X≤2) = P(X=0) + P(X=1) + P(X=2)
Using the same formula as before:
P(X≤2) ≈ 0.0282 + 0.1211 + 0.2335
P(X≤2) ≈ 0.3828
So the probability that at most two out of 10 customers will prefer the deluxe model is approximately 38.28%.