Question

the state lottery board is examining the machine that randomly picks the lottery numbers. on each trial, the machine outputs a ball with through 9 on it. (the ball is then replaced in the machine.) the lottery board tested the machine for 200 trials and got the following result

outcome	0	1	2	3	4	5	6	7	8	9

answer the following. round your answers to the nearest thousandths.
(a) from these results, compute the experimental probability of getting a 8.
(b) assuming that the machine is fair, compute the theoretical probability of getting a 8.
(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.

Explanation:

Part (a)

Step1: Recall experimental probability formula

Experimental probability of an event is the number of times the event occurs divided by the total number of trials. The formula is $P(\text{event}) = \frac{\text{Number of times event occurs}}{\text{Total number of trials}}$.

Step2: Identify values for event "getting an 8"

From the table, the number of trials where 8 occurred is 23, and the total number of trials is 200 (since $24 + 7 + 23 + 20 + 22 + 22 + 21 + 16 + 23 + 22 = 200$).

Step3: Calculate experimental probability

Using the formula, $P(8) = \frac{23}{200}$. Calculating this, $\frac{23}{200}= 0.115$.

Step1: Recall theoretical probability formula

For a fair machine (each outcome is equally likely), the theoretical probability of an event is the number of favorable outcomes divided by the total number of possible outcomes. Here, the possible outcomes are 0 - 9, so there are 10 possible outcomes, and only 1 favorable outcome (getting an 8).

Step2: Calculate theoretical probability

The formula is $P(\text{event})=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$. So, $P(8)=\frac{1}{10} = 0.1$.

• The first option is incorrect because a smaller number of trials means more variability, so experimental probability is less likely to be close to theoretical.
• The second option is incorrect because with a large number of trials, experimental probability can get close to theoretical (by the Law of Large Numbers).
• The third option is correct as the Law of Large Numbers states that as the number of trials increases, the experimental probability approaches the theoretical probability.

Answer:

0.115

Part (b)