If $$ A, B $$ are two events of the sample space of a random experiment and $$ P(A)=0.7 $$ $$ , P(B)=0.3, P(A \cap B)=0.2 $$, find : $$ \begin{array}{l} ext { 1) The probability of occurrence of the event } A ext { only. } \ ext { (2 } P(B) \ ext { B } P(A \cup B)\end{array} $$

Question

If $$ A, B $$ are two events of the sample space of a random experiment and $$ P(A)=0.7 $$ $$ , P(B)=0.3, P(A \cap B)=0.2 $$, find : $$ \begin{array}{l}	ext { 1) The probability of occurrence of the event } A 	ext { only. } \ 	ext { (2 } P(B) \ 	ext { B } P(A \cup B)\end{array} $$

UpStudy AI · Accepted Answer

To solve the problem, we will use the following probability formulas:

1. The probability of occurrence of event $$ A $$ only is given by:
   $$
   P(A \text{ only}) = P(A) - P(A \cap B)
   $$

2. The probability of event $$ B $$ is already given as $$ P(B) = 0.3 $$.

3. The probability of the union of events $$ A $$ and $$ B $$ is given by:
   $$
   P(A \cup B) = P(A) + P(B) - P(A \cap B)
   $$

Now, let's calculate each part step by step.

### Step 1: Calculate $$ P(A \text{ only}) $$

Using the formula:
$$
P(A \text{ only}) = P(A) - P(A \cap B)
$$
Substituting the known values:
$$
P(A \text{ only}) = 0.7 - 0.2 = 0.5
$$

### Step 2: State $$ P(B) $$

From the problem, we have:
$$
P(B) = 0.3
$$

### Step 3: Calculate $$ P(A \cup B) $$

Using the formula:
$$
P(A \cup B) = P(A) + P(B) - P(A \cap B)
$$
Substituting the known values:
$$
P(A \cup B) = 0.7 + 0.3 - 0.2 = 0.8
$$

### Summary of Results

1. The probability of occurrence of the event $$ A $$ only is $$ P(A \text{ only}) = 0.5 $$.
2. The probability of event $$ B $$ is $$ P(B) = 0.3 $$.
3. The probability of the union of events $$ A $$ and $$ B $$ is $$ P(A \cup B) = 0.8 $$.
- The probability of event $$ A $$ only is 0.5. - The probability of event $$ B $$ is 0.3. - The probability of either $$ A $$ or $$ B $$ occurring is 0.8.

If \( A, B \) are two events of the sample space of a random experiment and \( P(A)=0.7 \) \( , P(B)=0.3, P(A \cap B)=0.2 \), find : \( \begin{array}{l}\text { 1) The probability of occurrence of the event } A \text { only. } \\ \text { (2 } P(B) \\ \text { B } P(A \cup B)\end{array} \)

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