Chapter 10.1, Problem 29E

Solution

To calculate: The probability that John will practice his piano lesson.

The probability that John will practice his piano lesson is $0.64$ .

Given information:

If it rains tomorrow, the probability is 0.8 that John will practice his piano lesion. If it does not rain tomorrow, there is only a $0.4$ chance that John will practice. Suppose that the chance of rain tomorrow is $60\%$ .

Calculation:

John’s practice depends on the occurrence of rain, so place the related to raining on the left and have the branches considering piano practice on the right, like so:

  

The probability of raining in the form of the decimal number:

  $\begin{array}{l}P\left(\text{rain}\right)=60\%\\ =\frac{6}{100}\\ =0.06\end{array}$

Subtract $P\left(\text{practice}\right)$ considering the day it rains from $1$ to calculate the probability of John not practicing if it rains:

  $\begin{array}{l}P\left(\text{practice, if rain}\right)=0.8\\ P\left(\text{don't, if rain}\right)=1-0.8\\ =0.2\end{array}$

To calculate the probability of John not practicing, if it doesn’t, subtract $P\left(\text{practice, if no rain}\right)$ from $1:$

  $\begin{array}{l}P\left(\text{practice, if no rain}\right)=0.4\\ P\left(\text{don't, if no rain}\right)=1-0.4\\ =0.6\end{array}$

Multiply the probability of each event occurring alone:

  $\begin{array}{l}P\left(A\text{ and }B\right)=P\left(A\right)\cdot P\left(B\right)\\ P\left(\text{rain and practice}\right)=0.6\cdot 0.8\\ =0.48\\ P\left(\text{no rain and practice}\right)=0.4\cdot 0.4\\ =0.16\end{array}$

Now, calculate the sum of the probabilities of the events:

  $\begin{array}{l}P\left(A\text{ or }B\right)=P\left(A\right)+P\left(B\right)\\ P\left(\text{practice}\right)=0.48+0.16\\ =0.64\end{array}$