Question

You shuffle a deck of cards and then start turning them over one at a time. The first one is red. So is the second. And the third. In fact, you are surprised to get 10 red cards in a row. You start thinking, "The next one is due to be black!" a) Are you correct in thinking that there's a higher probability that the next card will be black than red? Explain. b) Is this an example of the Law of Large Numbers? Explain.

Short answer

a) Yes, the next card is more likely black. b) No, this is not an example of the Law of Large Numbers.

Step-by-step solution

Understanding the Problem

We have a standard deck of 52 cards, consisting of 26 red cards (hearts and diamonds) and 26 black cards (clubs and spades). You have already drawn 10 red cards in a row.

Assessing Probability After Drawing 10 Red Cards

Regardless of the previous draws, each draw from a well-shuffled deck is statistically independent. This means the probability of drawing a red or black card does not depend on previous draws. With 42 cards left, we calculate the probability for the next card. Since there are 16 red cards left and 26 red cards already drawn, the probability of drawing the next red card is \( \frac{16}{42} \) and the probability of drawing a black card is \( \frac{26}{42} \).

Evaluating Statement a

According to the probabilities, the likelihood of drawing a black card next is indeed \( \frac{26}{42} \), which is greater than the probability of drawing a red card \( \frac{16}{42} \). Hence, it is accurate to believe that the next card is more likely to be black than red.

Explaining the Law of Large Numbers

The Law of Large Numbers states that as more observations are collected, the average of these observations will converge to the expected value. However, the situation here (only 10 draws) doesn't exemplify a large enough sample to rely on this law. We are observing variance in a small sample size rather than approaching the expected average of half red and half black.

Evaluating Statement b

This is not an example of the Law of Large Numbers, as you are dealing with a short sequence of draws not large enough to apply this law. The number of draws is too small to expect convergence to an average.

Key concepts

Law of Large Numbers

The concept of the Law of Large Numbers is foundational in understanding probability and outcomes over many trials. This law states that as the number of trials or observations increases, the average of the results will tend to get closer to the expected value. In simpler terms, if you flip a coin thousands of times, you should see roughly an equal number of heads and tails. This is because the expected probability for each side of a fair coin is 0.5.

However, in the scenario with a deck of cards where only ten cards have been drawn, the Law of Large Numbers is not applicable. This is because ten draws represent a very small sample size. The law becomes relevant only in situations where there are many more observations allowing the probabilities to better reflect the expected outcomes. In probability exercises, understanding when this law is relevant helps to avoid misunderstandings about probability expectations in small trials.

Independence of Events

Understanding the independence of events in probability is crucial for accurately calculating outcomes. Events are considered independent if the outcome of one does not affect the outcome of another. In card games, each draw from a well-shuffled deck is typically an independent event, meaning the chance of drawing a red or black card doesn't rely on previous draws.

In the exercise with the 10 red cards, even though intuitively one might feel that a black card is due, the draws are independent. This means each card still has its probability based on the current composition of the deck, not past results. After 10 red cards are drawn, the deck has shifted to have 16 red and 26 black cards remaining, making the probability of drawing a black card higher at this point, but only due to the remaining deck's composition, not past events.

Card Game Probability

Probability in card games often involves understanding the makeup of the deck and how it changes with each draw. A standard deck consists of 52 cards, with 26 red cards and 26 black cards. Probability calculations start by considering these base odds and then adjust based on the cards left in the deck.

In the given example, after drawing 10 red cards, the probability shifts because there are fewer red cards left. The next card being black becomes more likely, not because of previous draws but due to the remaining card ratio — specifically, there are 16 red and 26 black cards left. Hence, probability at this stage is calculated as follows:

• Probability of drawing a red card: \( \frac{16}{42} \)
• Probability of drawing a black card: \( \frac{26}{42} \)

Understanding these dynamic probabilities helps in making accurate predictions in card games and avoiding common misconceptions, such as assuming past results influence future outcomes directly in independent event situations.