A drawer contains 8 brown socks and 4 black socks. A sock is taken a random from the drawer, its colour noted and replaced. This is done three times. If X is the discrete variable, the number of brown socks taken. Draw the probability distribution of X.
Can't draw here. P(3 red socks) = (8/12)^3
how bout u post the hole answer to the question next time fool.
To draw the probability distribution of X, we need to determine the probability of each possible outcome for the number of brown socks taken.
Let's denote the event of drawing a brown sock as "B", and the event of drawing a black sock as "K".
The possible outcomes for the number of brown socks taken in three draws are as follows:
0 brown socks: This means all three socks drawn are black.
BBB: P(BBB) = P(K) * P(K) * P(K) = (4/12) * (4/12) * (4/12) = 64/1728
= 1/27
1 brown sock: This means exactly one brown sock is drawn in the three draws, and the other two are black.
BKK: P(BKK) = P(B) * P(K) * P(K) + P(K) * P(B) * P(K) + P(K) * P(K) * P(B)
= (8/12) * (4/12) * (4/12) + (4/12) * (8/12) * (4/12) + (4/12) * (4/12) * (8/12)
= 384/1728 + 384/1728 + 384/1728
= 1152/1728
= 2/3
2 brown socks: This means exactly two brown socks are drawn in the three draws, and the other one is black.
BBK, BKB, KBB: P(BBK) = P(B) * P(B) * P(K) + P(B) * P(K) * P(B) + P(K) * P(B) * P(B)
= (8/12) * (8/12) * (4/12) + (8/12) * (4/12) * (8/12) + (4/12) * (8/12) * (8/12)
= 256/1728 + 256/1728 + 256/1728
= 768/1728
= 4/9
3 brown socks: This means all three socks drawn are brown.
BBB: P(BBB) = P(B) * P(B) * P(B) = (8/12) * (8/12) * (8/12) = 512/1728
= 8/27
Now we can summarize the probability distribution of X:
X = 0: P(X = 0) = 1/27
X = 1: P(X = 1) = 2/3
X = 2: P(X = 2) = 4/9
X = 3: P(X = 3) = 8/27
So, the probability distribution of X is as follows:
X | 0 | 1 | 2 | 3
--------------------------------------------
P(X) | 1/27 | 2/3 | 4/9 | 8/27
To draw the probability distribution of X, we need to find the probability of each possible outcome.
In this case, X represents the number of brown socks taken out of three socks. X can take on the values 0, 1, 2, or 3.
Let's calculate the probability for each value of X:
For X = 0 (no brown socks taken):
The probability of picking a black sock on the first draw is 4/12 (since there are 4 black socks out of a total of 12 socks). The same probability applies to the second and third draws since the sock is replaced each time. Therefore, the probability is (4/12) * (4/12) * (4/12) = 64/1728 = 1/27.
For X = 1 (one brown sock taken):
The probability of picking a brown sock on the first draw is 8/12, and the probability of picking a black sock on the second and third draws is 4/12 each. Therefore, the probability is (8/12) * (4/12) * (4/12) = 128/1728 = 8/108.
For X = 2 (two brown socks taken):
Similarly, the probability of picking a brown sock on the first two draws is (8/12) * (8/12), and the probability of picking a black sock on the third draw is 4/12. Therefore, the probability is (8/12) * (8/12) * (4/12) = 256/1728 = 16/108.
For X = 3 (all three socks are brown):
The probability of picking a brown sock on all three draws is (8/12) * (8/12) * (8/12) = 512/1728 = 32/108.
Now, we can summarize the probability distribution of X:
X = 0: 1/27
X = 1: 8/108
X = 2: 16/108
X = 3: 32/108
Please note that these probabilities are simplified fractions. If you prefer decimal values, you can simplify them further.