$$ 1 \leftarrow $$ The probability of rain tomorrow is $$ \frac{3}{5} $$. If it rains tomorrow, the probability that John will practice his cello is $$ \frac{4}{5} $$. If it does not rain tomorrow, the probability that John will practice his cello is $$ \frac{1}{2} $$. What is the probability that John will practice his cello tomorrow? $$ P $$ (John will practice) $$ =\square $$ (Type an integer or a simplified fraction.)

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UpStudy AI · Accepted Answer

1. Define the events:
   - Let $$ R $$ be the event that it rains.
   - Let $$ C $$ be the event that John practices his cello.

2. Given probabilities:
   - $$ P(R) = \frac{3}{5} $$
   - $$ P(\text{not } R) = 1 - \frac{3}{5} = \frac{2}{5} $$
   - $$ P(C \mid R) = \frac{4}{5} $$
   - $$ P(C \mid \text{not } R) = \frac{1}{2} $$

3. Use the Law of Total Probability:
   $$
   P(C) = P(C \mid R)P(R) + P(C \mid \text{not } R)P(\text{not } R)
   $$

4. Substitute the given probabilities:
   $$
   P(C) = \frac{4}{5} \cdot \frac{3}{5} + \frac{1}{2} \cdot \frac{2}{5}
   $$

5. Calculate each term:
   - First term: $$\frac{4}{5} \cdot \frac{3}{5} = \frac{12}{25}$$
   - Second term: $$\frac{1}{2} \cdot \frac{2}{5} = \frac{2}{10} = \frac{1}{5}$$

6. Sum the terms:
   $$
   P(C) = \frac{12}{25} + \frac{1}{5} = \frac{12}{25} + \frac{5}{25} = \frac{17}{25}
   $$

Thus, the probability that John will practice his cello tomorrow is $$ \frac{17}{25} $$.
The probability that John will practice his cello tomorrow is $$ \frac{17}{25} $$.

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