probability help

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mnava18

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hello , im trying to do some probability problems in a text and ive ran into some confusion. this section is about using the addition rule.Im hoping somone on here can clarify this as its frustrating the heck out of me lol. i thought i had a pretty good handle on this section but this question confused the heck out of me when i saw the answer in the back of the book.


the question asked is

Age |18-21 || 22-29 || 30-39|| 40-49 || 50-59|| 60 + |
Responded | 73 || 255 || 245 || 136 || 138 || 202 |
Refused | 11 || 20 || 33 || 16 || 27 || 49 |

what is the probability that the selected person responded or is in the 18-21 age bracket

so according to the book , this is the answer

P(person responded or is in 18-21 backet)= 1049/1205 + 11/1205= 1060/1205 = 0.880

now , what has me confused is this. you can be both responded and in the 18-21 bracket, so this is not disjoint. but in the books answer , they count this problem as disjoint. shouldnt the answer be

1060/1205-73/1205? it is possible for both those events to occur. i dont get why they didnt do this and made the -P(AandB) part of the addition rule 0.or disjoint
 
nvm , i figured it out . i noticed that in the book , they already remove the numbers that are doubles , so they dont need to subtract the not disjointed part , so thats why its 0.
 
I think part of the difficulty is due to the nature of the word "or." In logic and statistics, "or" is usually inclusive, such that "P or Q" means that just P, just Q, and both P and Q are all valid (i.e. true) outcomes. However, in the English language, "or" is usually exclusive, such that "Would you like coffee or tea?" expects an answer of either "coffee" or "tea" but not "both."

That said, you can look at the probabilities in a slightly different way which might help you understand the book's answer. You want to know the probability that a given participant is in the 18-21 age bracket or responded to the survey. Adding up the numbers, we see that there were 1205 total participants. Of those, 1049 people responded. Next, we'll look at participants in the 18-21 age bracket. Adding up those numbers, we see that 84 people were in the 18-21 bracket. But, as you noted, participants can be in both categories at the same time. How many people are in both? Looking at the table, 73. Now because we added up all the people in the 18-21 bracket and all the people who responded, we ended up counting those 73 people who are in both categories twice. So we need to subtract off 73 people. The resulting equation:

1049/1205 + 84/1205 - 73/1205 = 1049/1205 + 11/1205 = 1060/1205
 
ksdhart said:
I think part of the difficulty is due to the nature of the word "or." In logic and statistics, "or" is usually inclusive, such that "P or Q" means that just P, just Q, and both P and Q are all valid (i.e. true) outcomes. However, in the English language, "or" is usually exclusive, such that "Would you like coffee or tea?" expects an answer of either "coffee" or "tea" but not "both."

That said, you can look at the probabilities in a slightly different way which might help you understand the book's answer. You want to know the probability that a given participant is in the 18-21 age bracket or responded to the survey. Adding up the numbers, we see that there were 1205 total participants. Of those, 1049 people responded. Next, we'll look at participants in the 18-21 age bracket. Adding up those numbers, we see that 84 people were in the 18-21 bracket. But, as you noted, participants can be in both categories at the same time. How many people are in both? Looking at the table, 73. Now because we added up all the people in the 18-21 bracket and all the people who responded, we ended up counting those 73 people who are in both categories twice. So we need to subtract off 73 people. The resulting equation:

1049/1205 + 84/1205 - 73/1205 = 1049/1205 + 11/1205 = 1060/1205
Click to expand...

yes thats what i ended up getting as well . thanks for clarifying that. but theres also one more thing thats got me confused. when we use subtraction part of the addition rule , p(AandB). this is the same as P(A)*P(B) right? so this part can be multiplied out. because theres some examples ive seen were it doesnt match up right. for example. the probability that youll get a heart or a red king from a deck of cards. which is 13/52+2/52-1/52 = 7/26 . but looking at P(HeartandRedking) if we work it out , it becomes = 13/52*2/52 = 1/104. which the whole equations becomes 13/52+2/52-1/104= 29/104 which is a differnt answer. are we not supposed to multiply it out and just assume that we know its 1/52 because theres only 1 king of hearts in the deck? it confuses me because I would think that if we multiplied it out, we would get 1/52. but it doesnt come out as that.
 
This reminds me, when I took my introduction to proofs class as an undergraduate, a student was pretty angry to find out that even "either, or" in mathematics was inclusive, on a graded assignment they got back. In English it is often used passively as exclusive. A pretty heated argument (only the student was heated) took up about 10 minutes of class time.
 
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