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 alternate when drawing four cards one at a time without replacement is the sum of the two probabilities calculated in the step-by-step solution.

Step-by-step solution

Calculate the probability of starting with a red card

When you start, you have a 50% chance of picking a red card. To continue to alternate, for the next three draws, you would need to draw spade, red, spade. The probabilities of drawing these are \(\frac{13}{51}\), \(\frac{12}{50}\) and \(\frac{12}{49}\) respectively. Multiply these probabilities together with first pick of red card likelihood, \(\frac{26}{52}\), we get the total likelihood for this scenario. This will give us: \(\frac{26}{52} * \frac{13}{51} * \frac{12}{50} * \frac{12}{49}\)

Calculate the probability of starting with a spade

The alternative scenario is starting with a spade and then alternating. The probabilities of drawing these are \(\frac{26}{51}\), \(\frac{12}{50}\), \(\frac{25}{49}\) respectively for red, spade, red. Multiply these probabilities together with first pick of red card likelihood, \(\frac{13}{52}\), to give us: \(\frac{13}{52} * \frac{26}{51} * \frac{12}{50} * \frac{25}{49}\)

Add the probabilities of step 1 and step 2

At last, the final answer will be the sum of the probabilities of the two scenarios. So we add our results from Step 1 and Step 2. This way, we calculate the total probability of drawing an alternating sequence of red cards and spades.