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 500 trials and got the following results.

(a) from these results, compute the experimental probability of getting a 1.

(b) assuming that the machine is fair, compute the theoretical probability of getting a 1.

(c) assuming that the machine is fair, choose the statement below that is true:
the larger the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.
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.

Explanation:

Step1: Calculate experimental probability for part (a)

Experimental probability formula: \(P=\frac{\text{Number of favorable trials}}{\text{Total number of trials}}\).
For getting a \(1\), number of favorable trials \(n = 51\), total number of trials \(N=500\).
So \(P=\frac{51}{500}=0.102\).

Step2: Calculate theoretical probability for part (b)

Since the machine is fair, there are \(10\) possible outcomes (\(0 - 9\)).
Theoretical probability formula for an equally - likely event: \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
For getting a \(1\), number of favorable outcomes \(m = 1\), total number of outcomes \(k = 10\).
So \(P=\frac{1}{10}=0.1\).

Step3: Analyze the statement for part (c)

The law of large numbers states that as the number of trials \(n\) increases, the experimental probability \(P_{exp}\) approaches the theoretical probability \(P_{theo}\).

Answer:

(a) \(0.102\)
(b) \(0.1\)
(c) The larger the number of trials, the greater the likelihood that the experimental probability will be close to the theoretical probability.