Suppose that you roll a pair of dice 1000 times and get seven 350 times. Based on these results, what is the probability that the next roll results in seven?

Verified step by step guidance

1. Understand the problem: You rolled two dice 1000 times and observed the sum of seven occurring 350 times. You want to estimate the probability that the next roll will result in a sum of seven based on this data.
2. Recall that the empirical probability of an event is calculated by dividing the number of times the event occurred by the total number of trials. In this case, the event is rolling a sum of seven.
3. Set up the formula for empirical probability: \(P(7) = \frac{\text{Number of times sum 7 occurred}}{\text{Total number of rolls}}\).
4. Substitute the given values into the formula: \(P(7) = \frac{350}{1000}\).
5. Interpret the result as the estimated probability that the next roll will result in a sum of seven, based on the observed data.

Key Concepts

Theoretical Probability

Theoretical probability is the expected likelihood of an event based on all possible equally likely outcomes. For rolling two dice, the probability of getting a sum of seven is 6/36 or 1/6, since there are six combinations that result in seven out of 36 total outcomes.

Empirical Probability

Empirical probability is calculated from observed data or experiments, reflecting the frequency of an event occurring in practice. In this case, rolling seven 350 times out of 1000 trials gives an empirical probability of 350/1000 = 0.35.

Law of Large Numbers

The Law of Large Numbers states that as the number of trials increases, the empirical probability tends to get closer to the theoretical probability. Thus, with many rolls, the observed frequency of rolling a seven should approach 1/6, despite short-term fluctuations.