Question

The probability of getting a king when a card is selected at random from a standard deck of 52 playing cards is \(\frac{1}{13}\). a. Give a relative frequency interpretation of this probability. b. Express the probability as a decimal rounded to three decimal places. Then complete the following statement: If a card is selected at random, I would expect to see a king about_____ times in 1000 .

Short answer

a. In terms of frequency, we can expect to get a King about 7.69% of the time in a large number of draws from a standard deck. b. As a decimal, this probability is approximately 0.077. In 1000 draws, we would expect to see a King around 77 times.

Step-by-step solution

Understand the Probability

A standard deck of 52 cards contains 4 Kings. Hence the probability of getting a King on any draw is the favourable outcomes divided by total outcomes, that is, the ratio of the number of kings (4) to the total number of cards (52). So, the probability of getting a King when a card is chosen at random is \(\frac{4}{52} = \frac{1}{13}\).

Relative Frequency Interpretation

Part a asks for a relative frequency interpretation of this probability. In terms of relative frequency, this probability means that if we repeat the experiment of randomly drawing a card from the deck a large number of times, we would expect to get a King about \(\frac{1}{13}\) or approximately 7.69% of those times.

Decimal Interpretation

Part b asks to express the probability as a decimal rounded to three decimal places. Therefore, we calculate the decimal form of \(\frac{1}{13}\) which gives about 0.077.

Prediction in Larger Sample Size

Part b further asks to predict how often a King would be seen in 1000 draws. This can simply be calculated by multiplying the probability of getting a king by 1000. Hence, when a card is selected at random 1000 times, we should expect to see a King about \(1000 * 0.077 \approx 77\) times.

Key concepts

Relative Frequency Interpretation

When we talk about probability in card games, the relative frequency interpretation provides a practical way to understand what probability numbers actually mean in terms of outcomes. In essence, if a card game is played repeatedly, the relative frequency is the proportion of a certain outcome (like drawing a King) occurring over a large number of games or trials.

Let's put this into context with our given exercise. The probability of drawing a King from a standard deck of 52 cards is \(\frac{1}{13}\). The relative frequency interpretation suggests that if we were to repeat the action of selecting a card from the deck many, many times, we would expect to pull a King out of the deck approximately \(\frac{1}{13}\) or 7.69% of the time. This interpretation becomes more accurate as the number of trials increases. It is a powerful concept because it connects the abstract idea of probability with the real-world outcomes we can observe and count.

Decimal Probability

Probability can also be expressed in decimal form, which is often more intuitive for understanding and calculation purposes, especially when dealing with predictions or expected occurrences over multiple trials. Converting a fraction to a decimal can be done simply by dividing the numerator by the denominator.

In the case of getting a King from a deck of 52 cards, the fractional probability is \(\frac{1}{13}\), which when converted into decimal form is approximately 0.077, rounded to three decimal places. This decimal representation reveals a more immediate grasp of the likelihood of the event – that there is a little under an 8% chance of drawing a King at any given draw. This conversion is of particular importance when you are trying to calculate the expected occurrences over a set number of trials, like predicting how often a King will appear in 1000 draws.

Expected Value

Expected value is a core concept in probability that represents the average result or outcome one can expect if an experiment is repeated many times. It takes into account all possible outcomes and the probabilities of those outcomes. This concept is crucial when assessing games of chance and can help us predict winnings or losses over time.

In our exercise, when predicting the number of times we would see a King in 1000 card drawings, we multiply the probability of drawing a King, in decimal form (0.077), by the number of draws (1000). This gives us an expected value of 77 Kings. Thus, if someone were to draw cards from the deck 1000 times, on average, they can expect to draw a King about 77 times. It's important to remember that the expected value is an average, so in practice, the actual number might be slightly more or less, but over time the average will trend towards the expected value.