Question

# Consider a sample space S, and two events A and B such that P$A ∩ B$ = 0.2, P$A ∪ B$ = 0.6, P$B ∪ ̄A$ = 0.8 $a$ [0.5 points] Calculate P $A$. $b$ [0.5 points] Calculation P $B$

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## Answer to a math question Consider a sample space S, and two events A and B such that P$A ∩ B$ = 0.2, P$A ∪ B$ = 0.6, P$B ∪ ̄A$ = 0.8 $a$ [0.5 points] Calculate P $A$. $b$ [0.5 points] Calculation P $B$

Clarabelle
4.7
To solve this problem, we can use the concept of set theory and probability.

Step 1: Let's break down the given information using the properties of set theory and probability.

Given:
P$A ∩ B$ = 0.2
P$A ∪ B$ = 0.6
P$B ∪ ̄A$ = 0.8

Step 2: We can use the formula for the probability of the union of two events to find P$A ∪ B$.

P$A ∪ B$ = P$A$ + P$B$ - P$A ∩ B$

Given that P$A ∪ B$ = 0.6 and P$A ∩ B$ = 0.2, we can substitute these values in the equation and solve for P$A ∪ B$:

0.6 = P$A$ + P$B$ - 0.2

Step 3: Rearrange the equation to solve for P$A$:

P$A$ = 0.6 - P$B$ + 0.2

Step 4: Now, we need to find P$B ∪ ̄A$. The complement of event A is denoted by ̄A, which represents all the outcomes in the sample space S that are not in A.

Using the formula for the probability of the union of two events, we have:

P$B ∪ ̄A$ = P$B$ + P$̄A$ - P$B ∩ ̄A$

We can simplify this equation to:

P$B ∪ ̄A$ = P$B$ + P$̄A$ - P$B ∩ ̄A$ = 0.8

Step 5: Since we want to find P$A$ and P$B$, we need to eliminate P$̄A$ and P$B ∩ ̄A$. To do this, we can use the complement rule and rewrite P$B ∪ ̄A$ in terms of P$A ∪ B$ as follows:

P$B ∪ ̄A$ = 1 - P$A ∪ B$

Substituting the given value of P$A ∪ B$ = 0.6, we have:

0.8 = 1 - 0.6

Step 6: Now we can solve for P$B$:

0.8 = 1 - P$A ∪ B$ + P$B$ = 0.4 + P$B$

P$B$ = 0.8 - 0.4 = 0.4

Step 7: Finally, substitute the value of P$B$ into the equation for P$A$ that we obtained earlier:

P$A$ = 0.6 - P$B$ + 0.2 = 0.6 - 0.4 + 0.2 = 0.4

$a$ P$A$ = 0.4
$b$ P$B$ = 0.4
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