Alice has two bags of marbles: Bag A contains 8 red and 3 blue marbles, and Bag B contains 5 red and 7 blue marbles. She randoml
1 answer:
It might help to draw out a probability tree. Check out the diagram below. The first two branches at the top represent bags A and B. The probability of picking either bag is 1/2 assuming each bag is equally likely to be picked.
If we go with bag A, then the probability of getting blue is 3/11 since there are 3 blue out of 8+3 = 11 total. The probability of getting bag A and a blue marble is (1/2)*(3/11) = 3/22 which I've marked in the diagram as well.
If we go with bag B, then the probability of getting blue is 7/12 since there are 7 blue out of 5+7 = 12 total. The probability of getting bag B and blue is (1/2)*(7/12) = 7/24
Add up the results found getting
P(blue) = P(blue & bag A) + P(blue & bag B)
P(blue) = 3/22 + 7/24
P(blue) = 36/264 + 77/264
P(blue) = 113/264
This is shown in the diagram as well.
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Recall that conditional probability in general is defined as follows
P(A given B) = P(A and B)/P(B)
In this problem, we'll have
P(bag B given blue) = P(bag B and blue)/P(blue)
Therefore, we can say...
P(bag B given blue) = P(bag B and blue)/P(blue)
P(bag B given blue) = (7/24)/(113/264)
P(bag B given blue) = (7/24)*(264/113)
P(bag B given blue) = (7*264)/(24*113)
P(bag B given blue) = (7*24*11)/(24*113)
P(bag B given blue) = (7*11)/(113)
P(bag B given blue) = 77/113
In short, the probability of getting bag B, given the marble is blue, is 77/113.
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<h3>Answer:</h3>
- g(x) has an axis of symmetry at x = 3
- g(x) is shifted right 3 units from the graph of f(x)
- g(x) is shifted up 4 units from the graph of f(x)
The vertex form of g(x) is ...
... g(x) = -(x -3)² +4
This is offset to the right by 3 and up by 4 from the parent function. (It is also first reflected across the x-axis.)
_____
<em>Vertex form</em>
You know the leading coefficient is -1 because that's what it is for x² in the given form. When you factor -1 from the first two terms, of the given form, you have ...
... g(x) = -1(x² -6x) -5
Half the x coefficient inside parentheses will be the constant in the squared binomial term, so that term is (x -3)². The constant in that square is +9, so adding that value inside and outside parentheses in g(x) gives ...
... g(x) = -1(x² -6x +9) -5 +9
... g(x) = -(x -3)² +4 . . . . . vertex form
_____
<em>About transformations</em>
g(x) = f(x -a) causes the graph of f(x) to be shifted "a" units to the right. For a function f(x) with an axis of symmetry at x=0, it moves the axis of symmetry to x=a.
g(x) = f(x) +a causes the graph of f(x) to be shifted "a" units up.
g(x) = -f(x) causes the graph of f(x) to be reflected across the x-axis.
Here, we have all three of these transformations. First is the reflection:
... f₁(x) = -f(x) = -x²
Then we have shifting to the right 3 units. (also moves the axis of symmetry)
... f₂(x) = f₁(x-3) = -(x -3)²
Finally, we have shifting up 4 units.
... g(x) = f₂(x) +4 = -(x -3)² +4
Answer:
The width of rectangle is 1 unit.
Step-by-step explanation:
Given:
The area of rectangle is 6 units and its width is 5 units less than the length.
So, to find the width of rectangle.
Let the length be and the width be
.
Now, to find the width we put the formula of area of rectangle:
Area = length × width
<em>On solving equation we get:</em>
Therefore, we take the value x=6 as it is positive value:
So, the width of rectangle:
Width =
=
=
Therefore, the width of rectangle is 1 unit.