Event S = customer chose a small car
Event L = customer chose a luxury car
Event B = customer chose a budget car
Event R = car breaks down
Event ~R = car does not break down
I'll draw a partial tree diagram, and let you fill out the rest.
See below.
The top row of branches represents S, L and B to represent "small", "luxury", and "budget" in that exact order.
The decimal numbers next to each branch represents the probability of randomly selecting that type of car
30 small + 22 luxury + 48 budget = 100 cars total
30/100 = 0.30 is the probability of selecting a small car
22/100 = 0.22 is the probability of selecting a luxury car
48/100 = 0.48 is the probability of selecting a budget car
Then for each of those 3 branches, we'll have 2 branches. That means we have 3*2 = 6 branches total.
The 2 additional branches represent "breaks down" and "does not break down", which I'll shorten to R and ~R respectively.
Keep in mind the probability P(R) changes depending on which car you selected.
For more information, check out the concept of "conditional probability".
P(R) = 0.12 if event S happens
P(R) = 0.09 if event L happens
P(R) = 0.40 if event B happens
their associated complementary probabilities are...
P(~R) = 1-0.12 = 0.88 if event S happens
P(~R) = 1-0.09 = 0.91 if event L happens
P(~R) = 1-0.40 = 0.60 if event B happens
Rephrased in slightly different notation
P(R given S) = 0.12
P(R given L) = 0.09
P(R given B) = 0.40
P(~R given S) = 0.88
P(~R given L) = 0.91
P(~R given B) = 0.60
Those 6 new decimal values will label each of the 6 new branches in the second row.
Here's the tree diagram so far
Next, we'll multiply values along the branches to find the answers we're after.
Along path S and path R directly underneath it, we multiply the values to get 0.30*0.12 = 0.036; the result is placed at the very bottom of this pathway
Another example: 0.30*0.88 = 0.264 when following the path from S to ~R
Here's the slightly more updated version of the tree diagram
I'll let you take over from here. Your task is to fill out the remaining empty boxes following similar steps shown above.
Once the tree diagram is completed, you'll have the answers needed.
For instance, the answer to P(small and breaks down) is the value 0.036 found earlier. This is because we followed the S pathway and R pathway.
There's a 3.6% chance of selecting a small car and having it break down.
The answer to P(small and does not break down) would be 0.264 since it follows the S to ~R pathway.
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Extra info:
Notice how
P(S and R) + P(S and ~R) = P(S)
0.036 + 0.264 = 0.30
This is not a coincidence.