To determine the best prediction for the number of times Jeremy will roll a number greater than 4, we can use probability.

Step 1: Identify the favorable outcomes

A fair 6-sided die has numbers $1, 2, 3, 4, 5, 6$. Numbers greater than 4 are $5$ and $6$. So, there are $2$ favorable outcomes.

Step 2: Calculate the probability

The probability of rolling a number greater than $4$ is:

$P(\text{greater than } 4) = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{2}{6} = \frac{1}{3}.$

Step 3: Predict the number of successes

Jeremy rolls the die $180$ times. The expected number of times a number greater than $4$ appears is the product of the total rolls and the probability:

$\text{Expected number} = 180 \times \frac{1}{3} = 60.$

Final Answer:

The best prediction is that Jeremy will roll a number greater than $4$ 60 times.

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Would you like further clarification or a detailed breakdown? Here are some related questions:

1. What is the probability of rolling exactly $60$ numbers greater than $4$ in $180$ rolls?
2. How does the expected value change if the die were biased?
3. What is the variance of the number of rolls where the outcome is greater than $4$?
4. How would the prediction change if Jeremy rolls a $12$-sided die instead?
5. What is the probability of rolling a number not greater than $4$ in one roll?

Tip: Expected values are the average outcome over a large number of trials, not guarantees of exact outcomes!