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I am going through a course on basic statistics and instructor presented a problem, with a solution. To me it looks like the problem does not have a solution at all, let alone the solution posted by instructor. Most likely I am wrong, could you point out where?

Below is the problem:

A station along Route 66 sells gas and postcards. The probability that a customer buys postcards is .4. The probability that a customer leaves without buying anything is .3. The probability that the customer buys both gas and postcards is .6. What is the probability that the customer buys gas? Answer: .5

The way I see it:

P(gas and postcard) = 0.6

P(postcard) = 0.4

P(gas or postcard) = 0.7

However, P(postcard) would be >= P(gas and postcard), because there's P(postcard and no gas)

What am I missing?

Artem Lebedev
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    Did you mean to say that the probability that a customer buys only postcards is $.4$? – lulu May 08 '22 at 14:41
  • Hint: With correct values in the task (which is suspicious as you've mentioned), what is $\Bbb P(A\cup B)-\Bbb P(A)+\Bbb P(A\cap B)$ equal to if $A$ is the event of buying a postcard and $B$ is the event of buying gas? – PinkyWay May 08 '22 at 14:42
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    I agree with you...I can't come up with any interpretation that makes sense. If you assume that the problem means "the probability that a customer buys cards, regardless of wether or not they buy gas, is $.4$ then this contradicts the assumption that the probability they buy both is $.6$ while, on the other hand, if you assume that it means "the probability that a customer buys cards only is $.4$" then the numbers simply don't add up. – lulu May 08 '22 at 14:48
  • @lulu Thanks for confirming my suspitions! – Artem Lebedev May 09 '22 at 18:00

1 Answers1

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A station along Route 66 sells gas and postcards. The probability that a customer buys postcards is .4. The probability that a customer leaves without buying anything is .3. The probability that the customer buys both gas and postcards is .6. What is the probability that the customer buys gas? Answer: .5

can be read two ways, both of which lead to negative probabilities. On is that the $0.4$ probability is for postcards with or without gas in which case the other information suggests the probability of buying postcards but not gas is an impossible $-0.2$, as in this diagram with the four possible purchase probabilities adding up to $1$:

enter image description here

while the other is that the $0.4$ probability is for postcards without gas in which case the other information suggests the probability of buying gas but not postcards is an impossible $-0.3$, as in this diagram

enter image description here

Another possibility is that the question has been transcribed wrongly with two of the numbers transposed and should have said something like:

A station along Route 66 sells gas and postcards. The probability that a customer buys postcards (with or without gas) is $\mathbf{0.6}$. The probability that a customer leaves without buying anything is $0.3$. The probability that the customer buys both gas and postcards is $\mathbf{0.4}$. What is the probability that the customer buys gas (with or without postcards)? Answer: $0.5$

which can be illustrated with this diagram where $0.2=0.6-0.4$ and $0.1=1-0.6-0.3$ and $0.5=0.4+0.1$

enter image description here

Henry
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  • Thanks a lot for such a detailed answer! This is exactly how I saw it. I am just copy paste the problem from the instructor's presentation, so I would assume that there is a typo there, like you suggested. – Artem Lebedev May 09 '22 at 17:52