QuestionFind $P(C)$ given $P(A$ or $C)=\frac{7}{10}$ , $P(A$ and $C)=\frac{2}{5}$ , and $2P(B$ and $C)=P(A$ and $C)$ . Also, find the probability that A, B, or C do NOT occur.

Studdy Solution

STEP 1

Assumptions1. We are dealing with probabilities, so all values will be between0 and1. . The events A, B, and C are not necessarily mutually exclusive or independent.
3. The notation $(A$ or $C)$ represents the probability of either event A or event C occurring.
4. The notation $(A$ and $C)$ represents the probability of both event A and event C occurring.
5. The notation $(B$ and $C)$ represents the probability of both event B and event C occurring.
6. The notation $(A$ or $B$ or $C)$ represents the probability of either event A, event B, or event C occurring.
7. The notation $(A'$ or $B'$ or $C')$ represents the probability of either event A, event B, or event C not occurring.

STEP 2

We are given that $2(B$ and $C)=(A$ and $C)$ . We can rearrange this equation to find $(B$ and $C)$ .
$(B \, and \, C) = \frac{(A \, and \, C)}{2}$

STEP 3

Plug in the given value for $(A$ and $C)$ to calculate $(B$ and $C)$ .
$(B \, and \, C) = \frac{2}{5} /2$

STEP 4

Calculate the value of $(B$ and $C)$ .
$(B \, and \, C) = \frac{2}{} /2 = \frac{1}{}$

STEP 5

We know that $(A$ or $C) =(A) +(C) -(A$ and $C)$ . We can rearrange this equation to find $(C)$ .
$(C) =(A \, or \, C) +(A \, and \, C) -(A)$

STEP 6

We don't know the value of $(A)$ , but we know that $(A) =(A$ or $C) -(C) -(A$ and $C)$ . So we can substitute this into the equation for $(C)$ .
$(C) =(A \, or \, C) +(A \, and \, C) - ((A \, or \, C) -(C) -(A \, and \, C))$

STEP 7

implify the equation to find $(C)$ .
$(C) =2(C)$

STEP 8

Divide both sides by2 to solve for $(C)$ .
$(C) =1$

STEP 9

The probability that events A, B or C do NOT take place is given by $(A'$ or $B'$ or $C') = -(A$ or $B$ or $C)$ .

STEP 10

We don't know the value of $(A$ or $B$ or $C)$ , but we know that $(A$ or $B$ or $C) =(A) +(B) +(C) -(A$ and $B) -(A$ and $C) -(B$ and $C) +(A$ and $B$ and $C)$ .

STEP 11

We don't know the values of $(A)$ , $(B)$ , $(A$ and $B)$ , or $(A$ and $B$ and $C)$ , but we know that $(A) =(A$ or $C) -(C) -(A$ and $C)$ , $(B) =(B$ or $C) -(C) -(B$ and $C)$ , $(A$ and $B) =(A) +(B) -(A$ or $B)$ , and $(A$ and $B$ and $C) =(A) +(B) +(C) -(A$ or $B$ or $C)$ .

STEP 12

Substitute these values into the equation for $(A'$ or $B'$ or $C')$ .
$(A' \, or \, B' \, or \, C') = - ((A \, or \, C) -(C) -(A \, and \, C) +(B \, or \, C) -(C) -(B \, and \, C) + -(A \, or \, B \, or \, C))$

STEP 13

implify the equation to find $(A'$ or $B'$ or $C')$ .
$(A' \, or \, B' \, or \, C') = -(A \, or \, C) -(B \, or \, C) +2(C) +2(A \, and \, C) +2(B \, and \, C) -$

STEP 14

implify further to find $(A'$ or $B'$ or $C')$ .
$(A' \, or \, B' \, or \, C') =2(C) +2(A \, and \, C) +2(B \, and \, C) -(A \, or \, C) -(B \, or \, C)$

STEP 15

Substitute the given values for $(C)$ , $(A$ and $C)$ , and $(B$ and $C)$ to calculate $(A'$ or $B'$ or $C')$ .
$(A' \, or \, B' \, or \, C') =2() +2(\frac{2}{5}) +2(\frac{}{5}) - \frac{7}{10} - \frac{}{5}$

STEP 16

Calculate the value of $(A'$ or $B'$ or $C')$ .
$(A' \, or \, B' \, or \, C') =2 + \frac{4}{5} + \frac{2}{5} - \frac{}{10} - \frac{}{5} = \frac{9}{10}$ The probability that events A, B or C do NOT take place is $\frac{9}{10}$ .

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