Question 1176457: If Betty is late for his geography class, she makes a greater effort to arrive on time for the next class and the probability that she is on time is 3/4, However, is she on time, she is liable to be less concerned abut punctuality for the next class and her probability of being on time drops to 1/2. Betty is on time Monday. Using a tree diagram find the probability that
a) Betty is on time on Wednesday
b) Betty is late on Thursday
thank you
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
I'll provide the answers first at the top of the page and explain each part down below
Answer to part a) 5/8
Answer to part b) 13/32
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Explanation for part a)
I'll define the following events
T = Betty is on time on Tuesday
W = Betty is on time on Wednesday
H = Betty is on time on Thursday
They have the following complementary events
T' = Betty is late on Tuesday
W' = Betty is late on Wednesday
H' = Betty is late on Thursday
Convention usually has the tick marks to indicate the opposite of an event.
This is one way to draw the probability tree diagram.
Each blue dot represents being on time, while the red dots are being late.
The use of "given" means that we assume that the event has happened.
So for example, saying "P(W given T)" is the same as "P(W) assuming event T has occurred".
Each green level represents a different day. We have Monday at the very top (she is on time for this day), then followed by Tuesday, and so on. Each time we arrive at a different day, we branch off in two directions to represent the different outcomes for the next day. The first time we branch off is from Monday and those two upper-most branches represent either being on time for Tuesday (left) or being late on Tuesday (right). Every time we go left we're on time. Every time we go right, we're late. You can swap the order if you want, and have left mean late, but be sure to be consistent. That way you can just look at the directions and see what's going on without having to study the notation closely.
If we follow the purple path as shown below
Then Betty is on time for Tuesday and she's on time for Wednesday. We've gone to the left two times.
We multiply the values found along the purple path to find that (1/2)*(1/2) = 1/4 is the probability she's on time for Wednesday and she was on time Tuesday.
In terms of notation, we would say P(W and T) = P(W given T)*P(T) = (1/2)*(1/2) = 1/4
This shortens to P(W and T) = 1/4.
This is not all of P(W) however since we have one more path to consider.
If we follow the orange path, then Betty is late for Tuesday, but on time for Wednesday.
We multiply the values we find along the orange path: (1/2)*(3/4) = 3/8
In terms of notation, P(W and T ' ) = P(W given T ' ) * P( T ' ) = (3/4)*(1/2) = 3/8
In short, P(W and T ' ) = 3/8
As you can see from the diagram, there are only two ways to get to a case where Betty is on time for Wednesday, and that's through the purple and orange paths. Those are mutually exclusive paths, meaning that she can only take one path. You cannot be both late and on time simultaneously for any given day.
This mutual exclusivity allows us to add the probabilities: (purple)+(orange) = 1/4 + 3/8 = 2/8 + 3/8 = 5/8
The probability she is on time for Wednesday is 5/8
In terms of the notation we set up
P(W) = 5/8
P(W') = 3/8
Note that P(W)+P(W') = 1.
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Explanation for part b)
Go back to the original tree diagram. Highlight these four different paths
The end result of each path is that we land on H' which represents being late on Thursday. There are twice as many landing spots compared to the previous day. You can think of it like a game of Plinko where the ball always lands on one of the H' spots.
For each path, multiply the fractions found along it.
purple = (1/2)*(1/2)*(1/2) = 1/8
green = (1/2)*(1/2)*(1/4) = 1/16
red = (1/2)*(3/4)*(1/2) = 3/16
blue = (1/2)*(1/4)*(1/4) = 1/16
Add up the results of each computation
purple + green + red + blue
1/8 + 1/16 + 3/16 + 1/32
4/32 + 2/32 + 6/32 + 1/32
(4+2+6+1)/32
13/32
The probability she is late on Thursday is 13/32
You can stop here if both explanations are satisfactory and you don't need further explanation. The next two sections discuss how to approach parts (a) and (b) without using a tree diagram.
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Non-visual approach to part a)
Skip this section if you prefer the visual approach to part (a)
Recall that,
T = Betty is on time on Tuesday
W = Betty is on time on Wednesday
H = Betty is on time on Thursday
T' = Betty is late on Tuesday
W' = Betty is late on Wednesday
H' = Betty is late on Thursday
For Tuesday:
P(T) = 1/2
P(T') = 1/2
since she was on time for Monday.
Then based on that, we know,
P(W given T) = 1/2
P(W' given T) = 1/2
Or if event T' happens, ie she's late on Tuesday, then,
P(W given T') = 3/4
P(W' given T') = 1/4
Next, we'll use the conditional probability definition
P(A given B) = P(A and B)/P(B)
which rearranges to
P(A and B) = P(A given B)*P(B)
To form these four equations
P(W and T) = P(W given T)*P(T) = (1/2)*(1/2) = 1/4
P(W' and T) = P(W' given T)*P(T) = (1/2)*(1/2) = 1/4
P(W and T') = P(W given T')*P(T') = (3/4)*(1/2) = 3/8
P(W' and T') = P(W' given T')*P(T') = (1/4)*(1/2) = 1/8
Then we'll use the law of total probability to compute P(W)
So,
P(W) = P(W and T) + P(W and T')
P(W) = 1/4 + 3/8
P(W) = 2/8 + 3/8
P(W) = 5/8
The probability that Betty is on time for Wednesday is 5/8
This helps confirm the result we got earlier when using the visual approach.
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Non-visual approach to part b)
Skip this section if you prefer the visual approach to part (b)
We'll use the same ideas discussed earlier. From part (a), we found,
P(W) = 5/8
which must mean
P(W') = 3/8
since P(W)+P(W') = 1.
If Betty is on time for Wednesday, then we have these conditional probabilities
P(H given W) = 1/2
P(H' given W) = 1/2
We can then further say
P(H and W) = P(H given W)*P(W) = (1/2)*(5/8) = 5/16
P(H' and W) = P(H' given W)*P(W) = (1/2)*(5/8) = 5/16
If Betty is late on Wednesday, then,
P(H given W') = 3/4
P(H' given W') = 1/4
So,
P(H and W') = P(H given W')*P(W') = (3/4)*(3/8) = 9/32
P(H' and W') = P(H' given W')*P(W') = (1/4)*(3/8) = 3/32
To summarize so far:
P(H and W) = 5/16
P(H' and W) = 5/16
P(H and W') = 9/32
P(H' and W') = 3/32
Then we'll apply the law of total probability
P(H') = P(H' and W) + P(H' and W')
P(H') = 5/16 + 3/32
P(H') = 10/32 + 3/32
P(H') = (10+3)/32
P(H') = 13/32
The probability Betty is late on Thursday is 13/32
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