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Suppose there is a $50$ percent chance it snows on Friday, $60$ percent chance it snows on Saturday, and $40$ percent chance it snows on both days. What is the probability that it snows on Friday, but not Saturday?

I have the following, but am not sure if it is correct:

$\textbf{Let:}$

  • $F =$ Event that it snows on Friday
  • $S =$ Event that it snows on Saturday

$\textbf{Want: }$

  • $P(FS^c)\\$

$\textbf{My Solution:}\\ P(FS^c) = P(S^c|F)P(F) = (0.4)(0.6)?$

The main problem I run into is if I used the correct value for $P(S^c|F)$. I got $0.4$ from the the fact that it does not rain on Saturday $100(1 - 0.6) = 40$ percent of the time. Should I be utilizing the fact that it rains on either day $60$ percent of the time, or did I approach the solution from the wrong angle?

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  • $\begingroup$ $0.4=P(F\cap S)=P(F)\cdot P(S|F)=0.5\cdot 0.8\ne 0.3=0.5\cdot 0.6=P(F)\cdot P(S) \Rightarrow P(S|F)\ne P(S) \\ \text{So, $F$ and $S$ are dependent}$. $\endgroup$
    – farruhota
    Feb 3, 2019 at 9:58

1 Answer 1

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To solve the problem, you can use the following identity: $$P(FS^c)=P(F)-P(FS)$$

Note that $P(S^c|F)=1-P(S|F) \ne 1-P(S)$.

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  • $\begingroup$ Thank you! That last line clarifies everything, since the events are not independent. $\endgroup$
    – darylnak
    Feb 3, 2019 at 7:11
  • $\begingroup$ Congratulations for the $101\,k$ reputation ! I am sorry to have missed the day on which you passed the $100$. Cheers. $\endgroup$ Feb 3, 2019 at 7:32
  • $\begingroup$ thank you, Claude. $\endgroup$ Feb 3, 2019 at 9:00

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