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Chapter 8: Problem 4

Cards are selected one at a time without replacement from a well-shuffled deck of 52 cards until an ace is drawn. Let \(X\) denote the random variable that gives the number of cards drawn. What values may \(X\) assume?

Short Answer

Expert verified
The random variable \(X\), which represents the number of cards drawn until an ace is drawn, can assume any integer value from 1 to 49, inclusive. In other words, \(X \in \{1, 2, 3, \dots, 48, 49\}\).

Step by step solution

01

Identify the minimum and maximum values of \(X\)

First, let's consider the best- and worst-case scenarios. - Best-case scenario: We draw an ace on the first draw. In this case, \(X=1\). - Worst-case scenario: We draw all 48 non-ace cards before finally drawing an ace on the 49th draw. In this case, \(X=49\). These two scenarios give us the minimum and maximum values of \(X\) respectively.
02

Determine the possible values of \(X\) in between the minimum and maximum values

Since we draw cards one at a time without replacement, the number of cards drawn until an ace is drawn will take on every integer value between the minimum and maximum values found in Step 1. So, the random variable \(X\) can take on every integer value in the range: \[X \in \{1, 2, 3, \dots, 48, 49\}\] This means that \(X\) can take on any integer value from 1 to 49, inclusive.

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