Ask a New Question
Menu

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

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
53
54
55
50
50
49
52
45
40
52
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
0
.
(b)From these results, compute the experimental probability of getting a
0
.
(c)Assuming that the machine is fair, choose the statement below that is true:
The smaller the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.
The experimental probability will never be very close to the theoretical probability, no matter the number of trials.
The larger the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.

1 answer

To solve the problem, let's break it down step by step.

(a) Theoretical Probability of Getting a 0

The machine outputs a ball with one of the digits from 0 to 9. Since there are 10 possible outcomes (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and assuming the machine is fair, the theoretical probability \(P(0)\) of getting a 0 is:

\[ P(0) = \frac{\text{Number of ways to get a 0}}{\text{Total number of outcomes}} = \frac{1}{10} = 0.1 \]

(b) Experimental Probability of Getting a 0

To compute the experimental probability of getting a 0, we need to take the results from the trials:

  • Number of trials resulting in a 0: 53
  • Total number of trials: 500

Thus, the experimental probability \(P_{exp}(0)\) of getting a 0 is:

\[ P_{exp}(0) = \frac{\text{Number of trials resulting in a 0}}{\text{Total number of trials}} = \frac{53}{500} = 0.106 \]

Rounding to the nearest thousandth gives:

\[ P_{exp}(0) \approx 0.106 \]

(c) True Statement

Now let's evaluate the statements about the relationship between the number of trials and the likelihood that the experimental probability will be close to the theoretical probability:

  1. The smaller the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.

    This statement is false. With fewer trials, random variation has a larger impact on the experimental probability.

  2. The experimental probability will never be very close to the theoretical probability, no matter the number of trials.

    This statement is also false. While there may be variability, with a large enough number of trials, the experimental probability tends to converge to the theoretical probability.

  3. The larger the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.

    This statement is true. As the number of trials increases, the Law of Large Numbers suggests that experimental probabilities will converge to the theoretical probability.

Summary of Answers

(a) \(P(0) = 0.1\)

(b) \(P_{exp}(0) \approx 0.106\)

(c) The larger the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.