The table shows the results of rolling a fair six-sided die. Complete parts (a) through (d) below.
| Outcome on Die | First 100 Trials | Second 100 Trials | 500 Trials |
|---|---|---|---|
| 1 | 18 | 15 | 84 |
| 2 | 14 | -19 | 88 |
| 3 | 19 | 14 | 86 |
| 4 | 21 | 16 | 79 |
| 5 | 14 | 12 | 89 |
| 6 | 14 | 24 | 74 |
(a) Using the table, find the empirical probability of rolling a 2 for the first 100 trials.
The empirical probability of rolling a 2 for the first 100 trials is $\square$ 0.14.
(Round to two decimal places as needed.)
(b) Using the table, find the empirical probability of rolling a 2 for the second 100 trials.
The empirical probability of rolling a 2 for second 100 trials is $\square$
(Round to two decimal places as needed.)
Answer (1)
Find the number of times '2' occurred in the second 100 trials: 19.
Calculate the empirical probability: 100 19 .
Convert to decimal: 0.19.
The empirical probability of rolling a 2 for the second 100 trials is 0.19 .
Explanation
Understand the problem The problem asks us to find the empirical probability of rolling a 2 in the second 100 trials of rolling a fair six-sided die. The table provides the number of times each outcome occurred in the second 100 trials.
Identify the number of occurrences According to the table, the outcome '2' occurred -19 times in the second 100 trials. However, the number of occurrences cannot be negative. It is likely a typo, so we will consider the absolute value, which is 19.
Calculate the empirical probability The empirical probability is calculated by dividing the number of times the event occurred by the total number of trials. In this case, the event is rolling a 2, which occurred 19 times, and the total number of trials is 100. Therefore, the empirical probability is: P ( 2 ) = 100 19
State the final answer Now, we convert the fraction to a decimal: P ( 2 ) = 100 19 = 0.19 The empirical probability of rolling a 2 in the second 100 trials is 0.19.
Examples
Empirical probability is useful in many real-world scenarios. For example, in quality control, you might test a batch of products and find that 2 out of 100 are defective. The empirical probability of a defective product is then 2/100 = 0.02. This helps you estimate the overall quality of your production process. Similarly, in marketing, you can track how many people click on an ad out of the total number of people who see it. This gives you the empirical probability of a click, which helps you assess the effectiveness of your advertising campaign.
