Three cards are randomly chosen, without replacement, from a standard deck of
52. The random variable X represents the number of hearts cards chosen.
Construct the probability distribution for X.
Sample space, S={0,1,2,3}
PDF:
-∞<X<0 P(X)=0
P(0)=39*38*37/3! /C(52,3) no heart
P(1)=13*39*38/2! /C(52,3) 1 heart, 2 cards of other suits
P(2)=13*12/2!*39 /C(52,3) 2 hearts, 1 card of other suits
P(3)=13*12*11/3! /C(52,3) 3 hearts
3<X<∞ P(X)=0
Make sure that the values of the discrete PDF add up to 1.
To construct the probability distribution for the random variable X, which represents the number of hearts cards chosen, we need to find the probability of each possible value of X.
Step 1: Determine the total number of ways to choose 3 cards from a standard deck of 52.
Since we're choosing 3 cards without replacement, the total number of ways to choose 3 cards from 52 is given by the combination formula:
C(n, k) = n! / (k!(n-k)!)
where n is the total number of items and k is the number of items being chosen. In this case, n = 52 (total number of cards) and k = 3 (number of cards being chosen).
C(52, 3) = 52! / (3!(52-3)!)
= 22,100
So, there are 22,100 ways to choose 3 cards from a standard deck of 52.
Step 2: Determine the number of ways to choose X hearts cards.
To find the number of ways to choose X hearts cards, we need to consider the possible values of X.
When X = 0, it means no hearts cards are chosen. There are 39 non-hearts cards to choose from, so the number of ways to choose 0 hearts cards is given by:
C(39, 3) = 39! / (3!(39-3)!)
= 9,029
When X = 1, it means exactly 1 hearts card is chosen. There are 13 hearts cards to choose from and 39 non-hearts cards to choose from, so the number of ways to choose 1 hearts card is given by:
C(13, 1) * C(39, 2) = (13! / (1!(13-1)!) * 39! / (2!(39-2)!))
= 13 * 741
= 9,633
When X = 2, it means exactly 2 hearts cards are chosen. There are 13 hearts cards to choose from and 39 non-hearts cards to choose from, so the number of ways to choose 2 hearts cards is given by:
C(13, 2) * C(39, 1) = (13! / (2!(13-2)!) * 39! / (1!(39-1)!))
= 78 * 39
= 3,042
When X = 3, it means all 3 hearts cards are chosen. There are 13 hearts cards to choose from, so the number of ways to choose all 3 hearts cards is given by:
C(13, 3) = 13! / (3!(13-3)!)
= 286
Step 3: Calculate the probabilities.
Now that we know the number of ways to choose each value of X, we can calculate the probabilities by dividing the number of ways by the total number of ways.
P(X = 0) = 9,029 / 22,100
P(X = 1) = 9,633 / 22,100
P(X = 2) = 3,042 / 22,100
P(X = 3) = 286 / 22,100
So, the probability distribution for X is as follows:
X | P(X)
-------------
0 | 9,029 / 22,100
1 | 9,633 / 22,100
2 | 3,042 / 22,100
3 | 286 / 22,100