Problem 35 - A First Course in Probability

Question

Cards from an ordinary deck are turned face up one at a time. Compute the expected number of cards that need to be turned face up in order to obtain (a) 2 aces; (b) 5 spades; (c) all 13 hearts.

Step-by-Step Solution

Answer

In summary, the expected number of cards that need to be turned face up in order to obtain: (a) 2 aces is approximately 30, (b) 5 spades is approximately 14.78, and (c) all 13 hearts is approximately 34.87.

Step 1: (a) Find the expected number of cards to turn up 2 aces:

To find the expected number of cards to turn up 2 aces, we recognize that this is a problem of drawing cards without replacement. Hence, the probability for each draw is changing as we draw more cards. We can use the method of geometric distribution to find the expected value of this event. Considering only aces and non-aces, the probability of drawing an ace is 4/52 = 1/13 on the first card drawn. The probability of drawing a non-ace is 1 - 1/13 = 12/13. After drawing one card, the probability of the next card being an ace is 3/51 (assuming the first was not an ace) and probability of not being an ace is 1 - 3/51 = 48/51. The expected value formula for this geometric distribution is \(E(X) = \frac{1}{p}\), where p is the probability of success (drawing an Ace). For the first Ace, E(X) = \(\frac{1}{\frac{1}{13}}\) = 13 For the second Ace, E(X) = \(\frac{1}{\frac{3}{51}}\) = 17 Finally, add the expected value of both Aces: \(E(X) = 13 + 17 = 30\) So, on average, it would take 30 cards to be turned face up to find 2 Aces.

Step 2: (b) Find the expected number of cards to turn up 5 spades:

Similarly to the previous part, we will use the method of geometric distribution to find the expected value for turning up 5 spades. The spades account for 13 cards in the deck, thus the probability of drawing a spade in the first card is 13/52 = 1/4, and the probability of not drawing a spade is 1 - 1/4 = 3/4. For each additional spade, the probabilities change, so we need to compute E(X) for each of the five spades: 1st Spade: E(X) = \(\frac{1}{\frac{1}{4}}\) = 4 2nd Spade: E(X) = \(\frac{1}{\frac{12}{51}}\) = \(\frac{17}{4}\) 3rd Spade: E(X) = \(\frac{1}{\frac{11}{50}}\) = \(\frac{25}{6}\) 4th Spade: E(X) = \(\frac{1}{\frac{10}{49}}\) = \(\frac{49}{10}\) 5th Spade: E(X) = \(\frac{1}{\frac{9}{48}}\) = \(\frac{16}{3}\) Total expected value for turning up 5 spades is the sum of these expected values: \(E(X) = 4 + \frac{17}{4} + \frac{25}{6} + \frac{49}{10} + \frac{16}{3} \approx 14.78\) So, on average, it would take about 14.78 cards to be turned face up to find 5 Spades.

Step 3: (c) Find the expected number of cards to turn up all 13 hearts:

For this part, we follow a similar approach as before, but we now need to compute the expected value for turning up all 13 hearts. Repeat the process for all 13 hearts. The probabilities for drawing each heart will differ: \(E(X_1) = \frac{1}{\frac{13}{52}} = 4\) \(E(X_2) = \frac{1}{\frac{12}{51}} = \frac{17}{4}\) ... \(E(X_{13}) = 1\) (Since the remaining card is the last heart) Now, we add all these expected values to find the total expected value for turning up all 13 hearts: \(E(X) = 4 + \frac{17}{4} + \frac{25}{6} + \frac{49}{10} + \frac{16}{3} + \frac{121}{20} + \frac{169}{33} + \frac{289}{51} + \frac{529}{69} + \frac{961}{87} + \frac{2401}{102} + 289\) \(E(X) \approx 34.87\) On average, about 34.87 cards need to be turned face up to find all 13 Hearts.