Question

A grocery bag can be classified as either paper or plastic. Suppose that 97​% of grocery bags are classified as plastic. ​(a) Two grocery bags are chosen at random. What is the probability that both grocery bags are plastic​? ​(b) Five grocery bags are chosen at random. What is the probability that all five grocery bags are plastic​? ​(c) What is the probability that at least one of five randomly selected grocery bags is paper​? Would it be unusual that at least one of five randomly selected grocery bags is paper​?

Answer 1

Answer with explanation:

Given : The probability of grocery bags are classified as plastic = 0.97

(a) f two bags are chosen at random.

Then , the probability that both grocery bags are plastic​ is given by :-

$^2C_2(0.97)^2(1-0.97)^0=0.9409$

(b) If five grocery bags are chosen at random.

Then , the probability that all five grocery bags are plastic​ is given by :-

$^5C_5(0.97)^5(1-0.97)^0\approx0.8587$

(c) The probability of getting paper = 1-0.97=0.03

The probability that at least one of five randomly selected grocery bags is paper :-

$P(x\geq1)=1-P(0)\\\\1-^5C_0(0.03)^0(0.97)^5\\\\=1-(0.03)^0(0.97)^5=0.1412659\approx0.14>0.05$

Thus , it would not be unusual that at least one of five randomly selected grocery bags is paper.

Answer 2

Probability is used to determine the chances of an event

• The probability that both grocery bags are plastic is 0.9409
• The probability that all five grocery bags are plastic is 0.8587
• The probability that at least one grocery bags is paper is 0.1413
• It would not be unusual that at least one of five randomly selected grocery bags is paper.

$p = 97\%$ -- chances that a bag is plastic

(a) Both plastics selected are plastic

This is calculated as:

$P(2) = p^2$

So, we have:

$P(2) = (97\%)^2$

$P(2) = 0.9409$

Hence, the probability that both grocery bags are plastic is 0.9409

(b) All five are plastic

This is calculated as:

$P(5) = p^5$

So, we have:

$P(5) = (97\%)^5$

$P(5) = 0.8587$

Hence, the probability that all five grocery bags are plastic is 0.8587

(c) At least one of the five is paper

The probability that none of the five is paper is the same as the probability that all five is plastic.

So:

$P(None) = 0.8587$

Using complement rule,

$P(At\ least\ 1) = 1 - P(None)$

So, we have

$P(At\ least\ 1) = 1 - 0.8587$

$P(At\ least\ 1) = 0.1413$

Hence, the probability that at least one grocery bags is paper is 0.1413

The above probability is greater than 0.05

Hence, it would not be unusual that at least one of five randomly selected grocery bags is paper.

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