The International Housewares Association (IHA) reported that...

The International Housewares Association (IHA) reported that $73 \%$ of all U.S. households have ceiling fans. In addition, $77 \%$ of all U.S. households have an outdoor grill. Suppose $60 \%$ of all U.S. households have both a ceiling fan and an outdoor grill. A U.S. household is randomly selected. a. What is the probability that the household has a ceiling fan or an outdoor grill? b. What is the probability that the household has neither a ceiling fan nor an outdoor grill? c. What is the probability that the household does not have a ceiling fan and does have an outdoor grill? d. What is the probability that the household does have a ceiling fan and does not have an outdoor grill?

The International Housewares Association (IHA) reported that $73 \%$ of all U.S. households have ceiling fans. In addition, $77 \%$ of all U.S. households have an outdoor grill. Suppose $60 \%$ of all U.S. households have both a ceiling fan and an outdoor grill. A U.S. household is randomly selected.
a. What is the probability that the household has a ceiling fan or an outdoor grill?
b. What is the probability that the household has neither a ceiling fan nor an outdoor grill?
c. What is the probability that the household does not have a ceiling fan and does have an outdoor grill?
d. What is the probability that the household does have a ceiling fan and does not have an outdoor grill?

Key Concepts

Set Difference in Probability

This concept involves finding the probability of outcomes that belong to one event and explicitly exclude the outcomes that are common with another event (i.e., the difference between two events). It is useful when calculating the probability of an event occurring without the occurrence of another related event.

Addition Rule for Probability

This rule is used to calculate the probability that at least one of two events occurs. It states that the probability of the union of two events is equal to the sum of their individual probabilities minus the probability of their intersection. This prevents double counting the outcomes where both events occur.

Intersection of Events

This concept refers to the situation where two events occur simultaneously. In probability, the intersection of events is denoted as A ? B and is essential for understanding overlapping probabilities and for applying the addition rule correctly.

Complement Rule

The complement rule is used to determine the probability that an event does not occur, which is calculated as one minus the probability that the event does occur. This concept is commonly applied when finding the probability of 'neither' scenario or 'not' an event.

Transcript

00:01 In question 18, it says we have the international housewheres association reported 73 % of all u .s.

00:06 Households have ceiling fans.

00:08 I'm just going to write probability of c, which is ceiling fans, to be 0 .73.

00:13 In addition, 77 % of all u .s.

00:16 Households have an outdoor grill.

00:17 We can call that g for grill, which is 0 .77.

00:21 Suppose 60 % of all u .s.

00:23 Households have both a ceiling fan and an outdoor grill.

00:26 So c and g is going to be 0 .60.

00:32 A u .s.

00:33 Household is randomly selected.

00:34 So we have a few questions to answer based off of this information.

00:37 Before i jump into this information, i'm going to take what was given to me, and i'm going to put that to a two -way table.

00:42 It's just a different way to organize the data, and it makes it pretty easy to answer some follow -up questions.

00:49 So we're going to say this left side is ceiling fan and not ceiling fan, grill, and not grill.

00:55 We know that ceiling fans represent.

00:57 0 .73, so non -silling fans is 0 .27, so that should be everybody as in 1 or 100%.

01:04 Grills should be 0 .77.

01:07 Non -grills, 0 .23 again, left or right, that does add up to 1 or 100%.

01:12 We're also told that silling fans and grills proportion is 0 .60, so that's the top left, where the c and the g intersect.

01:21 And at this point, we can just start filling in the box from left to write 0 .60 and 0 .13.

01:28 We'll add up to 0 .73.

01:30 So from top to bottom, 0 .60 plus 0 .17...

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