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The events $E$, $F$ and $P$ satisfy the constraints $(E \triangle F) = .7$, $P (E ∪ F) = .8$.

  • (A.) Calculate $P (E ∩ F)$, making sure that is uniquely determined.
  • (B.) Determine the range of possible values of $P (E)$.
  • (C.) There may be independent $E,F$ events satisfying the constraints?

So I have done (A.) $P (E ∩ F) = .1$ and $P(E)+P(F)=.9$

At this point, I think that for (B.) $P(E)$ should be $<.9$ but I can't find the low limit.

For point (C.) I know that $P (E ∩ F)=P(E)P(F)$ if independent events. So I think I have to replace it on $P (E ∪ F)=P(E)+P(F)-P (E ∩ F)$ and see if the values ould exist.

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For $(B)$, observe that $$ P(E) \geq P(E \cap F) = 0.1 $$ Also observe that you have over estimated the upper bound since $$ P(E) \leq P(E \cup F) = 0.8 $$ For $(C)$, if exist, then $$ \begin{cases} P(E) \cdot P(F) = 0.1 \\ P(E) + P(F) = 0.9 \end{cases} $$ Solving these equations, we can obtain one solution as $$ P(E) = \frac{0.9 + \sqrt{0.41}}{2},\quad P(F) = \frac{0.9 - \sqrt{0.41}}{2} \tag{$1$} $$ One example achieving these probabilities is a uniform random variable $X$ on $[0, 1]$. Let $E$ be the event that $\{X \leq \frac{0.9 + \sqrt{0.41}}{2}\}$ and $F$ be the event that $\{0.8 \geq X \geq \frac{0.7 + \sqrt{0.41}}{2}\}$. Then the probability of $E$ and $F$ satisfy $(1)$ and the conditions in the question.

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