Question

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 40 trials and got the following results. fill in the table below. round your answers to the nearest thousandth.

outcome	number of trials
1	3
2	4
3	5
4	6
5	8
6	2
7	3
8	3
9	3

(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 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.

Explanation:

Step1: Calculate total number of trials

The total number of trials is \(3 + 3+4 + 5+6 + 8+2+3+3+3=40\).

Step2: Calculate experimental probability of getting a 6 or 8

The number of times 6 or 8 occurred is \(8 + 3=11\). The experimental - probability \(P(E)\) is given by the formula \(P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\). So, \(P(E)=\frac{11}{40}=0.275\).

Step3: Calculate theoretical probability of getting a 6 or 8

Since there are 10 possible outcomes (0 - 9) and 2 favorable outcomes (6 and 8), the theoretical probability \(P(T)\) is \(P(T)=\frac{2}{10}=0.2\).

Step4: Analyze the relationship between experimental and theoretical probabilities

With a large number of trials, the difference between experimental and theoretical probabilities should be small.

Answer:

(a) Experimental probability: \(0.275\), Theoretical probability: \(0.2\)
(b) There might be a difference between the experimental and theoretical probabilities, but the difference should be small.