The state lottery board is examining the machine that randomly picks the lottery numbers. On each trial, the machine outputs a ball with one of the digits

Question

0
through
9
on it. (The ball is then replaced in the machine.) The lottery board tested the machine for
500
trials and got the following results.
Outcome
0
1
2
3
4
5
6
7
8
9
Number of Trials
51
52
47
58
37
50
46
55
57
47
Fill in the table below. Round your answers to the nearest thousandth.
(a)From these results, compute the experimental probability of getting a
5
or
8
.
(b)Assuming that the machine is fair, compute the theoretical probability of getting a
5
or
8
.
(c)Assuming that the machine is fair, choose the statement below that is true:
With a large number of trials, there must be no difference between the experimental and theoretical probabilities.
With a large number of trials, there might be a difference between the experimental and theoretical probabilities, but the difference should be small.
With a large number of trials, there must be a large difference between the experimental and theoretical probabilities.

Answer 1

To analyze the lottery board's results, we can fill in the calculations step by step.

(a) Experimental Probability of getting a 5 or an 8

1. Count the occurrences:

   • Number of trials where the outcome is 5: 50
   • Number of trials where the outcome is 8: 57

2. Total occurrences for 5 or 8: \[ \text{Total occurrences} = 50 + 57 = 107 \]

3. Total trials: \[ \text{Total trials} = 500 \]

4. Calculating the experimental probability: \[ P(5 \text{ or } 8) = \frac{\text{Total occurrences}}{\text{Total trials}} = \frac{107}{500} = 0.214 \] Rounding to the nearest thousandth: \[ P(5 \text{ or } 8) \approx 0.214 \]

(b) Theoretical Probability of getting a 5 or an 8

1. Calculating the theoretical probability: Since the machine is fair, each of the 10 digits (0 through 9) has an equal chance of being selected.

   \[ P(5) = \frac{1}{10} \quad \text{and} \quad P(8) = \frac{1}{10} \]

2. Combined probability for 5 or 8: \[ P(5 \text{ or } 8) = P(5) + P(8) = \frac{1}{10} + \frac{1}{10} = \frac{2}{10} = 0.2 \]

3. Rounding to the nearest thousandth: \[ P(5 \text{ or } 8) \approx 0.200 \]

(c) True Statement about Probabilities

The correct statement regarding the relationship between experimental and theoretical probabilities as the number of trials increases is: "With a large number of trials, there might be a difference between the experimental and theoretical probabilities, but the difference should be small."

Summary of Results:

• (a) Experimental Probability of getting a 5 or 8: \( \approx 0.214 \)
• (b) Theoretical Probability of getting a 5 or 8: \( \approx 0.200 \)
• (c) True statement about probabilities: With a large number of trials, there might be a difference between the experimental and theoretical probabilities, but the difference should be small.