6 There is a 25% chance that Claire will have to work tonight and cannot study for the big math test. If Claire studies, then she has an 80% chance of earning a good grade. If she does not study, she only has a 30% chance of earning a good grade. a. Draw a diagram to represent this situation. b. Calculate the probability of Claire earning a good grade on the math test. c.If Claire earned a good grade, what is the probability that she studied?

### Probability of Claire earning a good grade is \( 0.575 \).<br />### Probability that Claire studied given she earned a good grade is \( 0.6957 \).<br /><br /># Calculation Details:<br /><br />### Diagram:<br /> \[<br /> \begin{array}{c}<br /> \text{Claire works (0.25)} \rightarrow \text{Does not study (0.75)} \rightarrow \text{Good Grade (0.30)} \\<br /> \text{Claire works (0.25)} \rightarrow \text{Does not study (0.75)} \rightarrow \text{Not Good Grade (0.70)} \\<br /> \text{Claire does not work (0.75)} \rightarrow \text{Studies (0.80)} \rightarrow \text{Good Grade (0.80)} \\<br /> \text{Claire does not work (0.75)} \rightarrow \text{Studies (0.80)} \rightarrow \text{Not Good Grade (0.20)} \\<br /> \end{array}<br /> \]<br /><br />### Calculate Overall Probability of Good Grade:<br />1. Probability Claire works and earns a good grade: <br /> \( P(\text{Good Grade} | \text{Work}) = 0.25 \times 0.30 = 0.075 \)<br />2. Probability Claire does not work and earns a good grade:<br /> \( P(\text{Good Grade} | \text{No Work}) = 0.75 \times 0.80 = 0.60 \)<br />3. Total Probability \( P(\text{Good Grade}) \):<br /> \[<br /> P(\text{Good Grade}) = P(\text{Good Grade} | \text{Work}) + P(\text{Good Grade} | \text{No Work}) \\<br /> = 0.075 + 0.60 = 0.675<br /> \]<br /><br />### Calculate Conditional Probability (Bayes' Theorem):<br />\[<br />P(\text{Studied} | \text{Good Grade}) = \frac{P(\text{Good Grade} | \text{Studied}) \cdot P(\text{Studied})}{P(\text{Good Grade})}<br />\]<br />1. \( P(\text{Good Grade} | \text{Studied}) = 0.80 \)<br />2. \( P(\text{Studied}) = 0.75 \)<br />3. Therefore,<br />\[<br />P(\text{Studied} | \text{Good Grade}) = \frac{0.75 \times 0.80}{0.675} = 0.6957<br />\]