Question

the magazine mass marketing company has received 12 entries in its latest sweepstakes. they know that the probability of receiving a magazine subscription order with an entry form is 0.6. what is the probability that no more than 8 of the entry forms will include an order? round your answer to four decimal places.

Explanation:

Identify the parameters of the binomial distribution

Using the Binomial Distribution knowledge point

• Number of trials: \(n = 12\)
• Probability of success: \(p = 0.6\)
• Probability of failure: \(q = 1 - p = 0.4\)
• Target variable: \(X \sim \text{Binomial}(12, 0.6)\)

Set up the cumulative probability expression

Using the Binomial Distribution knowledge point
We need to find the probability that no more than 8 entry forms include an order:

$$ P(X \le 8) = \sum_{k=0}^{8} \binom{12}{k} (0.6)^k (0.4)^{12-k} $$

Calculate the complementary probability

Using the Binomial Distribution knowledge point

$$ P(X \le 8) = 1 - P(X \ge 9) = 1 - [P(X=9) + P(X=10) + P(X=11) + P(X=12)] $$

Compute individual probability terms

Using the Binomial Distribution knowledge point

$$
LATEXBLOCK0
$$

Sum the terms and subtract from 1

Using the Binomial Distribution knowledge point

$$
LATEXBLOCK1
$$

Answer:

The Magazine Mass Marketing Company has received 12 entries in its latest sweepstakes. They know that the probability of receiving a magazine subscription order with an entry form is 0.6. What is the probability that no more than 8 of the entry forms will include an order? Round your answer to four decimal places. <blank>0.7747</blank>