Old race car problem/puzzle

Question

This was in an old (1935) "brain teaser" book, and I can't figure it out how to solve it!

There's a car race during which the cars experience 4 different types of car trouble, e.g. flat tire, blown motor, etc. I can't recall exactly what they are let's say they are A, B, C, and D.

Also it is stated that: 95% of the cars experience trouble A, 85% of the cars experience trouble B, 75% of the cars experience trouble C, and 65% of the cars experience trouble D

Then the question that is asked is, what is the least percentage of cars which must have experienced all 4 car troubles?

Anybody have any ideas? Thanks!

Answer 1

The probability of having car troubles A, B, C, and D is given by $$P(A \cap B \cap C \cap D)=1-P((A \cap B \cap C \cap D)^c)=1-P(A^c \cup B^c \cup C^c \cup D^c).$$ Notice the upper bound on $P(A^c \cup B^c \cup C^c \cup D^c)$ (and therefore the lower bound on $P(A \cap B \cap C \cap D)$) is given by the case where the four sets $A^c, B^c, C^c,$ and $D^c$ are mutually exclusive (it helps to draw a picture to better understand this). In this case $$P(A^c \cup B^c \cup C^c \cup D^c)=P(A^c)+P(B^c)+P(C^c)+P(D^C)=.35+.25+.15+.05=0.8,$$ since $P(A^c)=.35, P(B^c)=.25, P(C^c)=.15,$ and $P(D^c)=.05$. Thus, the upper bound on the probability of having car troubles A, B, C, and D is given by $$P(A \cap B \cap C \cap D)=1-0.8=0.2.$$

The reason we cannot exactly calculate the probability $P(A \cap B \cap C \cap D)$ is that we do not have information about the probability of set intersections. We can, however, find an upper bound, which is given by the case where the sets do not overlap.