Question 1189737: Consider the experiment of rolling two dice and the following events:
A: ‘The sum of the dice is 9’ and B: ‘The first die is an even number’ and C: “The difference (absolute value) of the dice is 2”
Find (a) p(A and B) (HINT: You cannot assume these are independent events.)
(b) p(A or B)
(c) Are A and B mutually exclusive events? Explain.
(d) Are A and B independent events? Explain.
(e) Are B and C independent events? Explain.
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Define these three events
A = the sum of the dice is 9
B = the first die is an even number
C = the difference (absolute value) of the dice is 2
========================================================
Part (a)
If the first die is an even number, and the two dice add to 9, then we have these two possibilities
4+5 = 9
6+3 = 9
This is out of 36 total ways to roll two dice.
That leads to
P(A and B) = 2/36 = 1/18
========================================================
Part (b)
There are 4 ways to add to 9
3+6 = 9
4+5 = 9
5+4 = 9
6+3 = 9
Out of 36 total
So P(A) = 4/36 = 1/9
Half of the values on a die are even, so
P(B) = 1/2
Or you could note that there are 18 occurrences in which the first die is even out of 36 pairs total.
So 18/36 = 1/2.
From part (a) we found that
P(A and B) = 1/18
From those items, we then can say:
P(A or B) = P(A) + P(B) - P(A and B)
P(A or B) = 1/9 + 1/2 - 1/18
P(A or B) = 2/18 + 9/18 - 1/18
P(A or B) = (2+9-1)/18
P(A or B) = 10/18
P(A or B) = 5/9
========================================================
Part (c)
P(A and B) = 1/18 was found earlier
This value is nonzero which means A and B can happen at the same time.
Therefore the events A and B are NOT mutually exclusive
Mutually exclusive events are ones that cannot happen simultaneously (eg: having a coin land on heads and tails at the same time).
========================================================
Part (d)
If events A and B were independent, then
P(A and B) = P(A)*P(B)
would be a true equation
Recall that we found
P(A) = 1/9
P(B) = 1/2
So,
P(A)*P(B) = (1/9)*(1/2) = 1/18
which matches with P(A and B) = 1/18 found in part (a)
So P(A and B) = P(A)*P(B) is indeed a true equation.
Therefore, events A and B are independent.
========================================================
Part (e)
We could use the same trick as part (d), but I'll use a different route.
Let's lay out all the possible outcomes for event C
(1,3)
(2,4)
(3,1), (3,5)
(4,2), (4,6)
(5,3)
(6,4)
There are 8 dice rolls that have the face values 2 units apart.
We can say that P(C) = 8/36 = 2/9
If event B occurs, then that set is reduced to
(2,4)
(4,2), (4,6)
(6,4)
There are 4 items here out of 18 occurrences where the first die is even.
Notice that 4/18 = 2/9
In short, P(C) = 2/9 whether event B happens or not.
The presence of B doesn't affect P(C).
----------------------
Return to these pairs of values
(1,3)
(2,4)
(3,1), (3,5)
(4,2), (4,6)
(5,3)
(6,4)
If event C were guaranteed to happen, then these are the only outcomes possible.
P(B) = 4/8 = 1/2 because there are four instances of having an even number show up in the first slot out of eight items in that set.
This is identical to the P(B) = 1/2 value we found back in part (b)
So P(B) is not affected by C either.
Whether C happens or not, P(B) = 1/2.
----------------------
Overall, events B and C are independent. Neither event affects one another.
|
|
|
