Current learring objective: Notation for the Binomiat: $$ \mathrm{B}= $$ Binomial Probability Distribution Function Question 12 Prev Score: O of 4 points The owner of a small deli is trying to decide whether to discontinue selling magazines. He suspects that only $$ 18.6 \% $$ of his customers buy a magazine and he thinks that he might be abie to use the display space to sell something more profitable. Before making a final decision, he decides that for one day he will keep track of the number of customers that buy a magazine. Assuming his suspicion that $$ 18.6 \% $$ of his customers buy a magazine is correct, what is the probability that exactly 4 out of the first 14 customers buy a magazine? A 0.0699 B 0.2435 C 0.0240 D 0.1530

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UpStudy AI · Accepted Answer

To find the probability that exactly 4 out of the first 14 customers buy a magazine, we can use the Binomial Probability Distribution Function.

Given:
- $$ n = 14 $$ (total number of customers)
- $$ k = 4 $$ (number of customers buying a magazine)
- $$ p = 0.186 $$ (probability of a customer buying a magazine)

The Binomial Probability Distribution Function is given by:
$$ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} $$

Substitute the given values into the formula:
$$ P(X = 4) = \binom{14}{4} (0.186)^4 (1-0.186)^{14-4} $$

Now, we can calculate the probability using the Binomial Probability Distribution Function.
Calculate the value by following steps:
- step0: Calculate:
  $$ { }_{14}C_{4}\times 0.186^{4}\left(1-0.186\right)^{14-4}$$
- step1: Subtract the numbers:
  $$ { }_{14}C_{4}\times 0.186^{4}\times 0.814^{14-4}$$
- step2: Subtract the numbers:
  $$ { }_{14}C_{4}\times 0.186^{4}\times 0.814^{10}$$
- step3: Expand the expression:
  $$\frac{14!}{4!\times \left(14-4\right)!}\times 0.186^{4}\times 0.814^{10}$$
- step4: Subtract the numbers:
  $$\frac{14!}{4!\times 10!}\times 0.186^{4}\times 0.814^{10}$$
- step5: Reduce the fraction:
  $$\frac{14\times 13\times 12\times 11}{4!}\times 0.186^{4}\times 0.814^{10}$$
- step6: Multiply the terms:
  $$\frac{24024}{4!}\times 0.186^{4}\times 0.814^{10}$$
- step7: Divide the terms:
  $$1001\times 0.186^{4}\times 0.814^{10}$$
- step8: Convert the expressions:
  $$1001\left(\frac{93}{500}\right)^{4}\times 0.814^{10}$$
- step9: Convert the expressions:
  $$1001\left(\frac{93}{500}\right)^{4}\left(\frac{407}{500}\right)^{10}$$
- step10: Simplify:
  $$1001\times \frac{93^{4}}{500^{4}}\times \left(\frac{407}{500}\right)^{10}$$
- step11: Multiply the terms:
  $$\frac{1001\times 93^{4}}{500^{4}}\times \left(\frac{407}{500}\right)^{10}$$
- step12: Evaluate the power:
  $$\frac{1001\times 93^{4}}{500^{4}}\times \frac{407^{10}}{500^{10}}$$
- step13: Multiply the fractions:
  $$\frac{1001\times 93^{4}\times 407^{10}}{500^{4}\times 500^{10}}$$
- step14: Multiply:
  $$\frac{1001\times 93^{4}\times 407^{10}}{500^{14}}$$
The probability that exactly 4 out of the first 14 customers buy a magazine is approximately 0.1530.

Therefore, the correct answer is D. 0.1530.
The probability that exactly 4 out of the first 14 customers buy a magazine is approximately 0.1530.

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