An e-commerce website claims that 2% of people who visit the site make a purchas

Question

An e-commerce website claims that 2% of people who visit the site make a purchase. Complete parts a through d below based on a random sample of 15 people who visited the website.

a.       What is the probability that none of the people will make a purchase?

b.      What is the probability that less than 3 people will make a purchase?

c.       What is the probability that more than 1 person will make a purchase?

d.      Suppose that out of 15 customers, 3 made a purchase. What conclusions can be drawn about the e-commerce site

Explanation / Answer

Answer a. Using the binomial formula for binomial probability, we get

P(x) = n!/(n-x)!x! * p ^ x q^ (n-x)

Here n = 15, p = 0.02

1-p = 0.98

Probability that nobody will buy is nothing but p(x ) at 0

So using the formulae we get,

(15!/ (15-0)! * 0!)* (0.02)^0 * (0.98)^(15-0)

This is roughly equal to 0.7386 (rounded up to four decimal places)

Answer b. Probability that less than three people will make a purchase is nothing but

p(x) at 0, 1, and 2

For p(x) at 0, the value is 0.7386.

Similarly, p(x) at 1 = (15!/ (15-1)! * 1!)* (0.02)^1 * (0.98)^(15-1)

Which is equal to (15!/ (14)! * 1!)* (0.02)^1 * (0.98)^(14)

On solving, you get p(x) at 1 = 0.2261

Similarly, p(x) at 2 = (15!/ (15-2)! * 2!)* (0.02)^2 * (0.98)^(15-2)

(15!/ (13)! * 2!)* (0.02)^1 * (0.98)^(13)

=0.0324

Adding them in total you get value for less than 3 customers making the purchase

P(x) less than 3 = 0.7386 + 0.2261 + 0.0324

= 0.9970

Answer c. More than 1 person makes purchase.

Rather than checking this, you can check the probability for 1 or less than 1. And subtract it from 1 (as the total probability is always 1)

P(x >1) = 1- p(x=0)+p(x=1)

From step b, p(x=0) = 0.7386 + 0.2261

= 1- 0.9647

= 0.0353

Answer d. Assuming that 3 customers make the purchase in every 15 customers.

p(x= 3) = (15!/ (15-3)! * 3!)* (0.02)^3* (0.98)^(15-3)

(15!/ (15-3)! * 3!)* (0.02)^3 * (0.98)^(12)

The total value is 0.0029

Based on the probability outputs, it is not likely that 2% customers make purchase. As the probability is too low and contradicts the assumptions of company

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