at a schools open house, t - shirts and sweatshirts were sold. each item was purchased by either a student…

at a schools open house, t - shirts and sweatshirts were sold. each item was purchased by either a student or a parent. the two - way frequency table summarizes a random sample of 60 items sold that night.\n| | student | parent |\n|--|--|--|\n| t - shirt | 11 | 27 |\n| sweatshirt | 13 | 9 |\nlet sweatshirt be the event that an item randomly chosen from among those sold was a sweatshirt. let parent be the event that an item randomly chosen from among those sold was purchased by a parent. find the following probabilities. write your answers as decimals.\n(a) p(parent) =\n(b) p(sweatshirt and parent) =\n(c) p(sweatshirt | parent) =

at a schools open house, t - shirts and sweatshirts were sold. each item was purchased by either a student or a parent. the two - way frequency table summarizes a random sample of 60 items sold that night.\n| | student | parent |\n|--|--|--|\n| t - shirt | 11 | 27 |\n| sweatshirt | 13 | 9 |\nlet sweatshirt be the event that an item randomly chosen from among those sold was a sweatshirt. let parent be the event that an item randomly chosen from among those sold was purchased by a parent. find the following probabilities. write your answers as decimals.\n(a) p(parent) =\n(b) p(sweatshirt and parent) =\n(c) p(sweatshirt | parent) =

Answer

Explanation:

Step1: Calculate the total number of items purchased by parents.

From the table, the number of T-shirts purchased by parents is 27 and the number of sweatshirts purchased by parents is 9. $$ \text{Total items purchased by parents} = 27 + 9 = 36 $$ The total number of items sold is 60.

Step2: Calculate the probability P(parent).

This is the number of items purchased by parents divided by the total number of items. $$ P(\text{parent}) = \frac{\text{Number of items purchased by parents}}{\text{Total number of items}} = \frac{36}{60} = 0.6 $$

Step3: Calculate the probability P(sweatshirt and parent).

This is the number of sweatshirts purchased by parents divided by the total number of items. From the table, this number is 9. $$ P(\text{sweatshirt and parent}) = \frac{\text{Number of sweatshirts purchased by parents}}{\text{Total number of items}} = \frac{9}{60} = 0.15 $$

Step4: Calculate the conditional probability P(sweatshirt | parent).

This is the probability that an item is a sweatshirt given that it was purchased by a parent. It is the number of sweatshirts purchased by parents divided by the total number of items purchased by parents. $$ P(\text{sweatshirt} | \text{parent}) = \frac{\text{Number of sweatshirts purchased by parents}}{\text{Total number of items purchased by parents}} = \frac{9}{36} = 0.25 $$ Alternatively, using the formula for conditional probability: $$ P(\text{sweatshirt} | \text{parent}) = \frac{P(\text{sweatshirt and parent})}{P(\text{parent})} = \frac{0.15}{0.6} = 0.25 $$

Answer:

(a) $P(\text{parent}) = 0.6$ (b) $P(\text{sweatshirt and parent}) = 0.15$ (c) $P(\text{sweatshirt} | \text{parent}) = 0.25$