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Understanding probability through relative frequency involves framing the likelihood of an event as a ratio of occurrences over many trials. For example, in the exercise above, the probability of drawing a king from a standard deck of cards is given as \( \frac{1}{13} \) because there are 4 kings out of 52 cards. This fraction represents what we might expect in the long run: on average, if you were to repeatedly draw a single card at random from the deck, roughly 1 out of every 13 cards would be a king.

The relative frequency interpretation is therefore a way of predicting what might happen over many, many trials based on the proportion of possible successes. As more trials are conducted, the actual relative frequency (the actual proportion of outcomes) will tend to get closer and closer to the theoretical probability. This approach is especially useful in real-life scenarios where probabilities may not be fixed or apparent but can be estimated from historical data.

Probabilities are often given as fractions, but converting them to decimal form can make them easier to understand and use, particularly in calculations. To convert a fractional probability to a decimal, divide the numerator by the denominator. In our textbook example, the probability of drawing a king from a deck of cards is \( \frac{1}{13} \), which, when converted to decimal form, becomes approximately 0.0769.

For practical purposes, such as prediction or statistical analysis, decimals can be rounded to a fixed number of decimal places. Generally, probabilities are rounded to two or three decimal places, and the textbook solution rounds it to three places, obtaining 0.077. This decimal representation is much more convenient for calculations, such as finding expected frequencies over many trials, and is widely used in statistical software and probability calculations.

The expected value in probability is a measure of the average outcome that one can anticipate from a random event. To calculate the expected value, you multiply each possible outcome by its corresponding probability and then sum up all these products. For instance, if you want to find out how many times a king will appear over 1000 draws from a deck of cards, you can use the already converted decimal probability of 0.077.

Multiply this probability by the total number of trials, in this case, 1000: \[ 0.077 \times 1000 = 77 \]. This result tells us that we can 'expect' to see a king roughly 77 times if we perform this random draw 1000 times. While individual outcomes are random, and you might see a king more or less than 77 times in any given set of 1000 draws, the expected value gives you a sense of the long-term average if the experiment were repeated many times.