If 3 cards are drawn from a deck of 52 cards without replacement, find the expected number of clubs.
13/52 * 12/51 * 11/50
I got 75%
To find the expected number of clubs, we need to consider the probability of drawing 0, 1, 2, or 3 clubs. Let's break it down step by step.
Step 1: Determine the sample space
The sample space consists of all the possible outcomes when drawing 3 cards from a deck of 52 cards without replacement. The total number of outcomes in the sample space is given by 52 choose 3, denoted as C(52, 3).
Step 2: Calculate the probability for each outcome
We need to calculate the probability of drawing 0, 1, 2, or 3 clubs in these 3 cards.
Probability of drawing 0 clubs:
To get 0 clubs, we need to choose 3 cards from the 39 non-club cards out of 52. This can be calculated as C(39, 3)/C(52, 3).
Probability of drawing 1 club:
To get 1 club, we need to choose 1 club from the 13 clubs and 2 non-club cards from the remaining 39 non-club cards. This can be calculated as C(13, 1) * C(39, 2)/C(52, 3).
Probability of drawing 2 clubs:
To get 2 clubs, we need to choose 2 clubs from the 13 clubs and 1 non-club card from the remaining 39 non-club cards. This can be calculated as C(13, 2) * C(39, 1)/C(52, 3).
Probability of drawing 3 clubs:
To get 3 clubs, we need to choose all 3 cards from the 13 clubs. This can be calculated as C(13, 3)/C(52, 3).
Step 3: Calculate the expected value
The expected value of a random variable is the sum of each possible outcome multiplied by its corresponding probability.
Expected number of clubs = 0 * P(0 clubs) + 1 * P(1 club) + 2 * P(2 clubs) + 3 * P(3 clubs)
Expected number of clubs = 0 * (C(39, 3)/C(52, 3)) + 1 * (C(13, 1) * C(39, 2)/C(52, 3)) +
2 * (C(13, 2) * C(39, 1)/C(52, 3)) + 3 * (C(13, 3)/C(52, 3))
Now, you can calculate the expected number of clubs using the formulas provided above and the values of C(n, r) can be found using combinatorial formulas or by using a calculator or statistical software.