Answer:
4. A
5. B
6. C
Step-by-step explanation:
4.
In total there are 12 possible shakes.
The combinations are:
ice cream - pineapple - peanuts
ice cream - pineapple - granola
ice cream - strawberry - peanuts
ice cream - strawberry - granola
ice cream - banana - peanuts
ice cream - banana - granola
yogurt - pineapple - peanuts
yogurt - pineapple - granola
yogurt - strawberry - peanuts
yogurt - strawberry - granola
yogurt - banana - peanuts
yogurt - banana - granola
Therefore the chance is 1/12 for the combination ice cream, pineapple and granola.
5.
To find the total possible outcomes, count the last value in the tree diagram. There are 8 possible outcomes.
6.
Find the total number of balls by adding each of the colours.
1 + 3 + 8 = 12
There is 1 red ball. The probability of getting a red ball out of the total is 1/12.
The probability in words ordered in comparison to the fractions:
impossible-----unlikely------------likely-------------certain
0/12------------ 1/12--5/12--------7/12---11/12----------12/12
Since 1/12 is between 1/12 and 5/12, the possibility is unlikely.
is not a subspace of 
. A simple counter-example to the contrary: let 
with 
. However, scaling by -1 gives the vector 
and 
.