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
20
trials and got the following results.
Outcome
0
1
2
3
4
5
6
7
8
9
Number of Trials
0
4
0
1
1
1
6
2
2
3
Fill in the table below. Round your answers to the nearest thousandth.
(a)Assuming that the machine is fair, compute the theoretical probability of getting a
6
or
8
.
(b)From these results, compute the experimental probability of getting a
6
or
8
.
(c)Assuming that the machine is fair, choose the statement below that is true:
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.
With a large number of trials, there must be no difference between the experimental and theoretical probabilities.

Answer 1

To solve the problem, let’s start by breaking it down into parts:

(a) Theoretical Probability of Getting a 6 or 8

A fair machine has an equal probability of selecting each digit (0 to 9), which means each digit has a probability of:

\[ P(\text{each digit}) = \frac{1}{10} \]

The possible favorable outcomes for getting a 6 or an 8 are:

• 6
• 8

There are 2 favorable outcomes. Therefore, the theoretical probability of getting a 6 or an 8 is:

\[ P(6 \text{ or } 8) = P(6) + P(8) = \frac{1}{10} + \frac{1}{10} = \frac{2}{10} = 0.2 \]

Rounded to the nearest thousandth, the theoretical probability is:

\[ \boxed{0.200} \]

(b) Experimental Probability of Getting a 6 or 8

Next, we compute the experimental probability based on the results from the trials. The outcomes are given as follows:

• 0: 0 trials
• 1: 4 trials
• 2: 0 trials
• 3: 1 trial
• 4: 1 trial
• 5: 1 trial
• 6: 6 trials
• 7: 2 trials
• 8: 2 trials
• 9: 3 trials

Now, we find the total number of trials, which is already given as 20.

Now, we need the number of favorable outcomes (the count for 6 and 8):

• Number of trials for 6: 6
• Number of trials for 8: 2

Thus, the number of favorable outcomes for getting a 6 or an 8 is:

\[ \text{Total favorable outcomes for 6 or 8} = 6 + 2 = 8 \]

Now we can compute the experimental probability:

\[ P(6 \text{ or } 8)_{\text{experimental}} = \frac{\text{Number of favorable outcomes}}{\text{Total trials}} = \frac{8}{20} = 0.4 \]

Rounded to the nearest thousandth, the experimental probability is:

\[ \boxed{0.400} \]

(c) True Statement About Probabilities

Now we analyze the statement regarding the comparison of experimental and theoretical probabilities when the number of trials is large.

• "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."
• "With a large number of trials, there must be no difference between the experimental and theoretical probabilities."

The most accurate statement is the first one. While the law of large numbers suggests that experimental probabilities will converge towards theoretical probabilities as the number of trials increases, there may still be small differences due to random variation.

Thus, the correct choice is:

\[ \boxed{\text{With a large number of trials, there might be a difference between the experimental and theoretical probabilities, but the difference should be small.}} \]