A bag contains six marbles, of which four are red and two are blue. Suppose two marbles are chosen at random and X represents the number of red marbles in the sample.
Looking for the expected value of X.
Probability of 0 red = (2/6)(1/5) = 2/30
Probability of 1 red: 2*(4/6)(2/5) = 16/30
Probability of 2 red: (4/6)*(3/5) = 12/20
Expectation value = 0/30 + 16/30 + 24/30 = 4/3
Well, let's see. We have six marbles in total, so there are 6 choose 2 = 15 different pairs of marbles that could be chosen.
Now, let's consider the possibilities for X.
If we choose two red marbles, then X = 2. There are 4 choose 2 = 6 ways to choose two red marbles.
If we choose one red marble and one blue marble, then X = 1. There are 4 ways to choose a red marble and 2 ways to choose a blue marble, so a total of 4 x 2 = 8 ways to choose one red and one blue marble.
If we choose two blue marbles, then X = 0. There are 2 choose 2 = 1 way to choose two blue marbles.
So, out of the 15 possible pairs of marbles, we have 6 pairs with X = 2, 8 pairs with X = 1, and 1 pair with X = 0.
Now, let's calculate the expected value of X.
E(X) = (probability of X = 2) x 2 + (probability of X = 1) x 1 + (probability of X = 0) x 0
The probability of X = 2 is 6/15, because there are 6 pairs with X = 2 out of 15 total pairs.
The probability of X = 1 is 8/15, because there are 8 pairs with X = 1 out of 15 total pairs.
The probability of X = 0 is 1/15, because there is 1 pair with X = 0 out of 15 total pairs.
Plugging these values into the equation, we get:
E(X) = (6/15) x 2 + (8/15) x 1 + (1/15) x 0
Simplifying, we find:
E(X) = 12/15 + 8/15 + 0
E(X) = 20/15
And finally, simplifying further:
E(X) = 4/3
So the expected value of X is 4/3.
And remember, when it comes to marble math, it's all about expecting the unexpected!
To find the expected value of X, we need to calculate the probability of each possible outcome and multiply it by the value of X for that outcome.
Let's denote R and B as the events of drawing a red and blue marble, respectively.
The possible outcomes for selecting two marbles are:
1. RR (both red)
2. RB (red then blue)
3. BR (blue then red)
4. BB (both blue)
Let's calculate the probability of each outcome:
1. Probability of RR:
P(RR) = (4/6) * (3/5) = 12/30 = 2/5
2. Probability of RB:
P(RB) = (4/6) * (2/5) = 8/30 = 4/15
3. Probability of BR:
P(BR) = (2/6) * (4/5) = 8/30 = 4/15
4. Probability of BB:
P(BB) = (2/6) * (1/5) = 2/30 = 1/15
Now, let's calculate the expected value of X. X can take values 0, 1, and 2.
For X = 0:
P(X = 0) = P(BB) = 1/15 * 0 = 0
For X = 1:
P(X = 1) = P(RB) + P(BR) = 4/15 + 4/15 = 8/15
For X = 2:
P(X = 2) = P(RR) = 2/5
Now, let's calculate the expected value:
E(X) = 0 * P(X = 0) + 1 * P(X = 1) + 2 * P(X = 2)
= 0 + 1 * 8/15 + 2 * 2/5
= 8/15 + 4/5
= 8/15 + 12/15
= 20/15
= 4/3
Therefore, the expected value of X is 4/3, which can also be written as 1.33.
To find the expected value of X, we need to calculate the probability of each possible outcome and multiply it by the value of X for that outcome.
There are two possible outcomes for X:
- X = 0, which means both marbles drawn are blue.
- X = 1, which means one red marble and one blue marble are drawn.
- X = 2, which means both marbles drawn are red.
Let's calculate the probabilities for each outcome:
1. X = 0:
To calculate the probability, we need to find the probability of drawing two blue marbles.
The probability of drawing a blue marble on the first draw is 2/6 since there are 2 blue marbles out of 6 total marbles.
After the first marble is drawn, there are 5 marbles remaining, with 1 blue marble.
So the probability of drawing a blue marble on the second draw is 1/5.
Therefore, the probability of X = 0 is (2/6) * (1/5) = 1/15.
2. X = 1:
To calculate the probability, we need to find the probability of drawing one red marble and one blue marble.
The probability of drawing a red marble on the first draw is 4/6 since there are 4 red marbles out of 6 total marbles.
After the first marble is drawn, there are 5 marbles remaining, with 2 blue marbles.
So the probability of drawing a blue marble on the second draw is 2/5.
Therefore, the probability of X = 1 is (4/6) * (2/5) = 4/15.
3. X = 2:
To calculate the probability, we need to find the probability of drawing two red marbles.
The probability of drawing a red marble on the first draw is 4/6.
After the first marble is drawn, there are 5 marbles remaining, with 3 red marbles.
So the probability of drawing a red marble on the second draw is 3/5.
Therefore, the probability of X = 2 is (4/6) * (3/5) = 12/30 = 2/5.
Now we can calculate the expected value:
E(X) = (Value of X = 0) * (Probability of X = 0) + (Value of X = 1) * (Probability of X = 1) + (Value of X = 2) * (Probability of X = 2)
E(X) = 0 * (1/15) + 1 * (4/15) + 2 * (2/5)
= 0 + 4/15 + 4/5
= 4/15 + 12/15
= 16/15
Therefore, the expected value of X is 16/15 or approximately 1.07.