Question

From a well shuffled deck of ordinary playing cards, four cards are turned over one at a time without replacement. What is the probability that the spades and red cards alternate?

Short answer

The probability that the spades and red cards drawn from a well-shuffled deck of 52 cards alternate is the sum of the probabilities of the two scenarios which start with either a spade or a red card.

Step-by-step solution

Identify the possible scenarios

Two scenarios are in consideration here: starting with a spade or starting with a red card.

Calculate the probabilities for the first scenario

For the first scenario, where we start with a spade, the probabilities for the sequence of draws are: first draw a spade \(\frac{13}{52}\), second draw a red card \(\frac{26}{51}\), third draw a spade \(\frac{12}{50}\), fourth draw a red card \(\frac{25}{49}\). Multiply these probabilities together to find the probability of this scenario.

Calculate the probabilities for the second scenario

For the second scenario, where we start with a red card, the probabilities for the sequence of draws are: first draw a red card \(\frac{26}{52}\), second draw a spade \(\frac{13}{51}\), third draw a red card \(\frac{25}{50}\), fourth draw a spade \(\frac{12}{49}\). Multiply these probabilities together to find the probability of this scenario.

Sum of probabilities

The two scenarios are mutually exclusive (as they can't both occur at the same time), so we add the two probabilities together to find the total probability.

Key concepts

Card Probability Without Replacement

Understanding the probability of drawing cards without replacement from a standard deck is essential in various probability exercises. This concept involves calculating the likelihood of drawing a card or a sequence of cards, bearing in mind that each draw affects the subsequent probability due to the reduction in the total number of cards.

For instance, the probability of drawing a spade from a full deck is \(\frac{13}{52}\). Once a card is drawn and if it's not replaced, the deck has 51 cards left, altering the probability of the next draw. If the first card was a spade, there are now 12 spades left out of 51 cards. Therefore the probability of drawing another spade is now \(\frac{12}{51}\). The 'without replacement' condition ensures the calculation accounts for the changing composition of the deck after each draw, a critical factor when solving for sequential probability.

Alternate Sequence Probability

The probability of an alternate sequence of events, like drawing alternate suits from a card deck, requires us to consider multiple, sequential conditional probabilities. These scenarios often involve complex calculations because the outcome of previous events influences the probability of future events.

In the given exercise, we are interested in alternating sequences of spades and red cards. To assess this, we imagine two distinct paths the sequence can take: starting with a spade or beginning with a red card. Each path consists of a series of dependent events where the occurrence of one modifies the likelihood of the next. For example, if we start with a spade, the chance of drawing a red card next is affected by the fact that a spade was just removed from the deck. This sequence-specific approach allows us to accurately calculate the probability for the entire sequence, as demonstrated in Step 2 and Step 3 of the solution, using the fundamental rules of probability and the concept of conditional probability.

Mutually Exclusive Events

Mutually exclusive events are those that cannot happen at the same time. In probability theory, this concept is crucial for determining the likelihood of a particular outcome amongst several distinct possibilities.

In card probability problems, events like drawing a spade or drawing a heart on a single pick are mutually exclusive; a card cannot be both a spade and a heart simultaneously. This principle allows us to add the probabilities of each exclusive event for scenarios where one or the other occurs, but not both (as seen in Step 4 of the solution). It's important to note that this rule of addition applies only when dealing with mutually exclusive events. If events are not mutually exclusive, meaning they could potentially occur simultaneously, their probabilities should not be simply added together.