Question 559429: There is an interview where 100 people were interviewed and asked questions about if they heard a specific song on the radio television or both. 50 people listened to it on the radio, 30 people heard it on the television, and 20 people heard it on both. My question is: How and why does P(heard it on the television or radio) the same as P(heard it on the radio + heard it on the television - heard it on both)?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the answer depends on how the categories have been created.
if you separate people who have heard it on both radio and tv in a separate category, then what you get is the following:
radio DISABLED_event_only= 50
tv DISABLED_event_only= 30
both = 20
total people = 100
probability of someone hearing it on the radio = (50 + 20)/100 * 100% = 70%
probability of someone hearing it on the tv = (30 + 20)/100 * 100% = 50%
probability of someone hearing it on both = 20/100 * 100% = 20%
notice that the probability of someone hearing it on the radio includes the number of people who heard it on both, and the probability of someone hearing it on the tv also includes the number of people who heard it on both.
now, when you are given the probability of someone hearing it on radio and you are given the probability of someone hearing it on tv, and you want to find the number of people who heard it on the tv or on the radio, you would need to subtract the number of people who heard it on both to remove the double counting.
you would get:
number of people who heard it on radio or on tv = 70% + 50% - 20% = 100%
the subtraction of the probability of the number of people who heard it on both is necessary to remove the double counting.
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