Let's break down the problem step-by-step to understand if Mary was correct or incorrect.
### Step 1: Understanding the Simplified Probability
The experimental probability given is [tex]\(\frac{1}{6}\)[/tex]. This means that, when simplified, the ratio of occurrences to trials should be equal to [tex]\(\frac{1}{6}\)[/tex].
### Step 2: Finding the Number of Trials for the Simplified Probability
To find the correct number of trials that would give an experimental probability of [tex]\(\frac{1}{6}\)[/tex] with 9 occurrences, we can set up the following equation:
[tex]\[
\text{Number of trials} \times \frac{1}{6} = 9 \implies \text{Number of trials} = 9 \times 6 = 54
\][/tex]
So, there should have been 54 trials to obtain an experimental probability of [tex]\(\frac{1}{6}\)[/tex] with 9 occurrences.
### Step 3: Verification of Mary's Proposed Trials
Mary proposed that there were 45 trials. We need to check if this would result in the simplified experimental probability of [tex]\(\frac{1}{6}\)[/tex].
Calculate the experimental probability using 45 trials:
[tex]\[
\text{Experimental probability with 45 trials} = \frac{9 \text{ occurrences}}{45 \text{ trials}} = \frac{9}{45} = \frac{1}{5}
\][/tex]
### Step 4: Comparison with the Simplified Probability
Now, compare the experimental probability with the simplified probability of [tex]\(\frac{1}{6}\)[/tex]:
[tex]\[
\frac{1}{5} \neq \frac{1}{6}
\][/tex]
### Conclusion
Since [tex]\(\frac{1}{5}\)[/tex] (0.2) does not equal [tex]\(\frac{1}{6}\)[/tex] (approximately 0.1667), Mary is incorrect in stating that there were 45 trials. The correct number of trials should have been 54 to achieve the given simplified probability of [tex]\(\frac{1}{6}\)[/tex] with 9 occurrences. Hence, based on this analysis, we conclude that Mary's proposed trials lead to a different probability, making her statement incorrect.
