Question 1125388
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Define the two events:
A = person sees advertisement
B = person buys product

<img src = "https://i.imgur.com/MLq7WjT.png">


What is the probability that the customer purchase?


P(B) = probability of making purchase
P(B) = 0.31 because this is the total in the "purchase" column. This means 31% of people bought the product; therefore the chances of making a purchase is <font color=red>0.31 or 31%</font>

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What is the probability that the customer purchase if he sees the ads?


P(B|A) = probability of making purchase given they have seen the ad
P(B|A) = P(B and A)/P(B)
P(B|A) = 0.22/0.31  <font color=blue>The value 0.22 is in the "buys product" column and "sees ad" row</font>
P(B|A) = <font color=red>0.709677</font>
<font color=red>If the customer sees the ad, then the probability is approximately 0.709677</font>


Note how the probability changes based on the prior knowledge of seeing the advertisement.

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Are the events of “See the Ads” and “Purchase” independent? Show why or why not.


Since P(B) = 0.31 and P(B|A) = 0.709677, this means that P(B|A) = P(B) is not a true equation. Therefore, <font color=red>A and B are <b><u>not</u></b> independent events</font>. We consider them to be dependent events.


If A and B were independent, then both of the following equations would be true
P(A|B) = P(A)
P(B|A) = P(B)


Writing 
P(B|A) = P(B)
means that event A has no effect on event B, so the probabilities would not alter. However, P(B) changes once we know that event A happens.

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Are the ads effective? Explain 


<font color=red>Yes the advertising is effective</font>. At first, P(B) = 0.31 which is fairly low. Then P(B) jumps up to 0.709677 when we compute P(B|A) = 0.709677


Another way to think of P(B|A) is to break it down into "P(B) given that event A has occurred". So after we know the person saw the ad, the chances of them buying the product is about 71%, which is quite the dramatic increase compared to 31%. Spending money on advertising is a wise choice.
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