Assume that a deck of cards contains 4 aces, 5 other face cards, and 13 non-face cards, and that you randomly draw 3 cards. A random variable Z is defined to be 3 times the number of aces plus 2 times the number of other face cards drawn.
Complete the probability density function.
To find the probability density function of Z, we'll consider all possible values that Z can take and compute the probability of each.
Z can take values: 0 (no aces, no face cards), 2 (1 face card, no aces), 3 (1 ace, no other face cards), 4 (2 face cards, no aces), 6 (3 face cards or 2 aces), and 9 (3 aces). We now compute the probability for each case:
1. P(Z = 0): No aces or face cards.
We must draw all 3 cards from the 13 non-face cards: C(13,3)/C(22, 3) = 286/1540
2. P(Z = 2): One face card, no aces.
Choose 1 face card from the 5 available and 2 from the 13 non-face cards: [C(5, 1) * C(13, 2)]/C(22, 3) = 325/1540
3. P(Z = 3): One ace, no face cards.
Choose 1 ace from the 4 available and 2 from the 13 non-face cards: [C(4, 1) * C(13, 2)]/C(22, 3) = 260/1540
4. P(Z = 4): Two face cards, no aces.
Choose 2 face cards from the 5 available and 1 from the 13 non-face cards: [C(5, 2) * C(13, 1)] / C(22, 3) = 260/1540
5. P(Z = 6): Three face cards or 2 aces.
Three face cards: C(5, 3)/C(22, 3) = 10/1540
Two aces and one non-face card: [C(4, 2) * C(13, 1)]/C(22, 3) = 156/1540
Combined: (10 + 156)/1540 = 166/1540
6. P(Z = 9): Three aces.
Choose all 3 aces from the 4 available: C(4, 3)/C(22, 3) = 4/1540
So the probability density function of Z is:
P(Z) = {
9/1540, if Z = 0,
13/1540, if Z = 2,
26/1540, if Z = 3,
26/1540, if Z = 4,
83/1540, if Z = 6,
2/1540, if Z = 9,
0, otherwise
}
To find the probability density function (PDF), we need to calculate the probability of each possible value of the random variable Z.
Let's calculate the probabilities for each value of Z:
Z = 0:
This means no aces and no other face cards are drawn.
P(Z = 0) = P(no aces) * P(no other face cards)
= (10/22) * (9/21) [10 non-aces out of 22 total cards, then 9 other non-face cards out of the remaining 21 total cards]
= 45/154
Z = 3:
This means 1 ace and no other face cards are drawn.
P(Z = 3) = P(1 ace) * P(no other face cards)
= (4/22) * (9/21) [4 aces out of 22 total cards, then 9 other non-face cards out of the remaining 21 total cards]
= 36/154
Z = 6:
This means 2 aces and no other face cards are drawn.
P(Z = 6) = P(2 aces) * P(no other face cards)
= (4/22) * (8/21) [4 aces out of 22 total cards, then 8 other non-face cards out of the remaining 21 total cards]
= 32/154
Z = 9:
This means 3 aces and no other face cards are drawn.
P(Z = 9) = P(3 aces) * P(no other face cards)
= (4/22) * (7/21) [4 aces out of 22 total cards, then 7 other non-face cards out of the remaining 21 total cards]
= 28/154
Z = 8:
This means 0 aces and 4 other face cards are drawn.
P(Z = 8) = P(no aces) * P(4 other face cards)
= (10/22) * (5/21) [10 non-aces out of 22 total cards, then 5 other face cards out of the remaining 21 total cards]
= 50/154
Z = 10:
This means 1 ace and 4 other face cards are drawn.
P(Z = 10) = P(1 ace) * P(4 other face cards)
= (4/22) * (5/21) [4 aces out of 22 total cards, then 5 other face cards out of the remaining 21 total cards]
= 20/154
Z = 12:
This means 2 aces and 4 other face cards are drawn.
P(Z = 12) = P(2 aces) * P(4 other face cards)
= (4/22) * (4/21) [4 aces out of 22 total cards, then 4 other face cards out of the remaining 21 total cards]
= 16/154
Note: Z cannot be any higher since the maximum value of Z is 12 (3 aces * 3 + 2 other face cards * 3 = 9 + 6 = 15). All other values of Z are not possible within the given deck of cards.
Putting it all together, the probability density function (PDF) for the random variable Z is:
P(Z = 0) = 45/154
P(Z = 3) = 36/154
P(Z = 6) = 32/154
P(Z = 9) = 28/154
P(Z = 8) = 50/154
P(Z = 10) = 20/154
P(Z = 12) = 16/154
The PDF is a function that assigns probabilities to each possible value of a random variable. We have calculated the probabilities for each possible value of Z based on the given deck of cards and the conditions.
To complete the probability density function (PDF) for the given scenario, we need to calculate the probability of each possible outcome of the random variable Z.
The random variable Z is defined as follows:
Z = 3 * [number of aces drawn] + 2 * [number of other face cards drawn]
Let's consider the possible values of the random variable Z:
1. Z can be 0 if no aces or other face cards are drawn.
2. Z can be 3 if 1 ace is drawn and no other face cards are drawn.
3. Z can be 5 if no aces are drawn and 1 other face card is drawn.
4. Z can be 6 if 2 aces are drawn and no other face cards are drawn.
5. Z can be 8 if 1 ace and 1 other face card are drawn.
6. Z can be 9 if no aces are drawn and 2 other face cards are drawn.
7. Z can be 12 if 3 aces are drawn and no other face cards are drawn.
8. Z can be 14 if 2 aces and 1 other face card are drawn.
9. Z can be 15 if 1 ace and 2 other face cards are drawn.
10. Z can be 18 if 3 aces and 1 other face card are drawn.
11. Z can be 20 if 2 aces and 2 other face cards are drawn.
12. Z can be 21 if 1 ace and 3 other face cards are drawn.
13. Z can be 24 if 3 aces and 2 other face cards are drawn.
To calculate the probability of each possible value of Z, we need to find the probability for each case and add them up.
1. Probability of Z = 0:
- The probability of drawing no aces or other face cards is: (13/22) * (12/21) * (11/20).
- Multiply this probability by 3^0 * 2^0 to get the final probability.
2. Probability of Z = 3:
- The probability of drawing 1 ace and no other face cards is: (4/22) * (13/21) * (12/20).
- Multiply this probability by 3^1 * 2^0.
3. Probability of Z = 5:
- The probability of drawing no aces and 1 other face card is: (18/22) * (5/21) * (13/20).
- Multiply this probability by 3^0 * 2^1.
4. Probability of Z = 6:
- The probability of drawing 2 aces and no other face cards is: (4/22) * (3/21) * (13/20).
- Multiply this probability by 3^2 * 2^0.
5. Probability of Z = 8:
- The probability of drawing 1 ace and 1 other face card is: (4/22) * (5/21) * (12/20).
- Multiply this probability by 3^1 * 2^1.
6. Probability of Z = 9:
- The probability of drawing no aces and 2 other face cards is: (18/22) * (17/21) * (5/20).
- Multiply this probability by 3^0 * 2^2.
7. Probability of Z = 12:
- The probability of drawing 3 aces and no other face cards is: (4/22) * (3/21) * (2/20).
- Multiply this probability by 3^3 * 2^0.
8. Probability of Z = 14:
- The probability of drawing 2 aces and 1 other face card is: (4/22) * (3/21) * (5/20).
- Multiply this probability by 3^2 * 2^1.
9. Probability of Z = 15:
- The probability of drawing 1 ace and 2 other face cards is: (4/22) * (18/21) * (17/20).
- Multiply this probability by 3^1 * 2^2.
10. Probability of Z = 18:
- The probability of drawing 3 aces and 1 other face card is: (4/22) * (3/21) * (1/20).
- Multiply this probability by 3^3 * 2^1.
11. Probability of Z = 20:
- The probability of drawing 2 aces and 2 other face cards is: (4/22) * (3/21) * (1/20).
- Multiply this probability by 3^2 * 2^2.
12. Probability of Z = 21:
- The probability of drawing 1 ace and 3 other face cards is: (4/22) * (18/21) * (1/20).
- Multiply this probability by 3^1 * 2^3.
13. Probability of Z = 24:
- The probability of drawing 3 aces and 2 other face cards is: (4/22) * (3/21) * (1/20).
- Multiply this probability by 3^3 * 2^2.
Once you calculate all these probabilities, normalize them by dividing each probability by the total sum of probabilities. This will give you the complete probability density function for the random variable Z.