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 1000 trials and got the following results.

outcome	0	1	2	3	4	5	6	7	8	9
number of trials	102	122	95	95	87	90	94	98	98	119

answer the following. round your answers to the nearest thousandths.
(a) from these results, compute the experimental probability of getting an odd number.
(b) assuming that the machine is fair, compute the theoretical probability of getting an odd number.
(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 no difference between the experimental and theoretical probabilities.
with a large number of trials, there must be a large difference between the experimental and theoretical probabilities.

Explanation:

Step1: Identify odd - numbered outcomes

Odd - numbered outcomes are 1, 3, 5, 7, 9.

Step2: Calculate the sum of trials for odd - numbered outcomes

Sum of trials for odd - numbered outcomes = 122+95+90+98+119 = 524.

Step3: Calculate the experimental probability

Experimental probability \(P_{e}=\frac{\text{Number of favorable trials}}{\text{Total number of trials}}\). Total number of trials = 1000. So \(P_{e}=\frac{524}{1000}=0.524\).

Step4: Calculate the theoretical probability

There are 10 possible outcomes (0 - 9) and 5 odd - numbered outcomes (1, 3, 5, 7, 9). Theoretical probability \(P_{t}=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}=\frac{5}{10}=0.5\).

Step5: Analyze the relationship between experimental and theoretical probabilities

With a large number of trials, the experimental probability approaches the theoretical probability, but there might still be a small difference.

Answer:

(a) 0.524
(b) 0.5
(c) With a large number of trials, there might be a difference between the experimental and theoretical probabilities, but the difference should be small.