# Suppose that when the weather is rainy the probability that Lily goes to her outdoor baseball practice is 0.1, but when it's not raining the probability is 0.9. The probability it will rain tomorrow is 0.3. Determine the probability that She will go to her baseball practice tomorrow.

Probability
Suppose that when the weather is rainy the probability that Lily goes to her outdoor baseball practice is 0.1, but when it's not raining the probability is 0.9. The probability it will rain tomorrow is 0.3. Determine the probability that She will go to her baseball practice tomorrow.

2020-11-11
Let A denote the event that Lily goes to practice, and let B denote the event that rains. Then
$$\displaystyle{P}{\left({A}\right)}={P}{\left({A}{\mid}{B}\right)}{P}{\left({B}\right)}+{P}{\left({A}{\mid}{B}^{{c}}\right)}{P}{\left({B}^{{c}}\right)}$$
(B* is the complementary event of B — here it is the event that it does not rain). Also,
$$\displaystyle{P}{\left({B}\right)}={0.3}\Rightarrow{P}{\left({B}^{{c}}\right)}={1}-{P}{\left({B}\right)}={0.7}$$
Also, we are given that
$$\displaystyle{P}{\left({A}{\mid}{B}\right)}={0.1}$$
$$\displaystyle{P}{\left({A}{\mid}{B}^{{c}}\right)}={0.9}$$
Therefore,
$$\displaystyle{P}{\left({A}\right)}={0.1}\cdot{0.3}+{0.9}\cdot{0.7}={0.66}$$